WASHINGTON — Over a breakfast of bacon and eggs in his California district last week, Representative Kevin McCarthy, the House Republican leader, tried to calm the nerves of a small group of longtime donors who raised questions about the extremists in his conference.

Some of the lawmakers’ comments and views may seem outrageous, he told the donors gathered at a restaurant overlooking a golf course. But on visits to congressional districts, he said, it was clear to him that the right-wing lawmakers were merely doing what the voters who sent them to Washington wanted.

Hours later, Mr. McCarthy did what the fringe wanted: He endorsed the woman running in Wyoming’s Republican primary to oust the far right’s archnemesis, Representative Liz Cheney, a former member of his leadership team who has earned pariah status in her party by speaking out against former President Donald J. Trump and the deadly attack on the Capitol that he helped inspire with lies of a stolen election.

The day exemplified the tightrope Mr. McCarthy is walking as he plots a path to become the next speaker of the House. Even as he courts the mainstream elements of his party, he has defended Republicans who have called the Jan. 6 riot a righteous cause. And he sided against a member of his own conference in throwing his support behind the Wyoming primary challenger, Harriet Hageman, whose central message is that Ms. Cheney should be ousted for breaking with Mr. Trump and daring to investigate the most brutal attack on the Capitol in centuries.

If Republicans win the majority this fall, Mr. McCarthy will need the support of the whole party, including the big donors who fund it, a dwindling number of center-right traditionalists and a larger group of quiet conservatives.

But he will also need the smaller but more powerful faction of extremist members who are aligned with Mr. Trump and want to define their party in his image. They are skeptical of the brand of mainstream Republicanism that propelled Mr. McCarthy’s rise; some are openly hostile to it.

So Mr. McCarthy has been engaging in a series of political contortions to try to secure a foothold in a party that has shifted under his feet, catering to a group that may ultimately be his undoing. In doing so, he has both empowered the hard-right fringe and tethered his fate to it, helping to solidify its dominance in today’s Republican Party.

“There was probably a time when it made sense to have someone like Kevin McCarthy, but we need new leadership in the House,” said Joe Kent, a square-jawed former Special Forces officer who is trying to unseat Representative Jaime Herrera Beutler in Washington, one of 10 Republicans who voted to impeach Mr. Trump after the Capitol attack. “He’s used to a different era.”

He added, “Our job is to obstruct and impeach, not to cut any deals.”

One Republican House member who backs Mr. McCarthy, who insisted on anonymity to discuss his predicament candidly without fear of a backlash from colleagues or constituents, said that as hard as Mr. McCarthy was working to maintain control, some in the party were so extreme that his position had become all but untenable.

Last week, the former Fox Business personality Lou Dobbs, who carries sway with Mr. Trump, was musing on a podcast with one of the right’s most pro-Trump voices, Representative Matt Gaetz of Florida, that Mr. McCarthy was a “RINO” — one of the former president’s favorite insults for people he considers to be “Republican in name only” — who had no business being speaker.

“The party needs strength,” Mr. Dobbs told Mr. Gaetz, who is under federal investigation for possible sex trafficking of a minor. “It needs vision. It needs energy, vibrancy and new blood in leadership. It’s that simple.”

With his political future in many ways out of his hands, Mr. McCarthy is leaving little to chance. His sunny disposition, prodigious fund-raising and ability to remember the names of the children of every House Republican are well known among his colleagues. And he has toiled to transform himself from a glad-handing, business-backed Republican from Bakersfield, Calif., into a credible leader of House’s far right, even as he assures donors that he remains an ally who knows how to navigate a debt ceiling increase and bills to fund the government.

Still, he faces unique troubles, including the prospect that he could face a subpoena from the House committee investigating the Jan. 6 attack, which regards him as a crucial witness because of his conversations with Mr. Trump during and after the riot. He has been consulting with William A. Burck, a prominent Washington lawyer, about how to navigate the investigation.

For now, Mr. McCarthy is spending ample time trying to influence Mr. Trump. He speaks to or visits the former president about every other week, most of the time with his top political aide, Brian Jack, who served as the White House political director under Mr. Trump.

Current and former aides to Mr. Trump describe Mr. McCarthy’s relationship with the former president as cordial but lacking in any loyalty. They are not in lock step on which candidates to support in the midterm elections, and Mr. McCarthy knows he ultimately has limited influence over Mr. Trump’s endorsements. That has not stopped the House leader from trying.

For instance, he sought to persuade Mr. Trump to stay out of Representative Rodney Davis’s re-election race in Illinois. Instead, the former president heeded the advice of Representative Marjorie Taylor Greene of Georgia to endorse Representative Mary Miller, who was thrown into the same district as Mr. Davis by the state’s Democratic gerrymander. Ms. Miller made an approving reference to Hitler at a rally last year in Washington.

Mr. McCarthy has had more success privately urging Mr. Trump not to get involved in the re-election campaign of Representative David Valadao of California, who voted for Mr. Trump’s impeachment. Mr. Valadao represents the most heavily Democratic district held by any Republican in Congress, Mr. McCarthy has explained to Mr. Trump, so endorsing a more conservative candidate could cost the party the seat.

So far, Mr. Trump has remained silent. But his aides said that is likely driven as much by the fact that no serious challenge has emerged as it is by the persuasiveness of Mr. McCarthy’s case.

There was a time when Mr. McCarthy appeared to be ready to break more decisively with the former president. In the immediate wake of the Jan. 6 assault, he called for Mr. Trump to be censured, stating on the House floor that he “bears responsibility” for the riot. He also called for an independent investigation of what had happened.

But later, Mr. McCarthy visited the former president at his Florida resort to make amends and enlist his help in the midterm elections, and then he fought the creation of an inquiry at every turn.

Last month, he defended the Republican National Committee after it passed a resolution to censure Ms. Cheney and the other Republican member of the Jan. 6 committee, Representative Adam Kinzinger of Illinois; the resolution said they were involved in the “persecution of ordinary citizens engaged in legitimate political discourse.” In contrast, Senator Mitch McConnell of Kentucky, the Republican leader, castigated the party.

In private talks to donors, Mr. McCarthy often does not mention Mr. Trump as he makes his aggressive pitch about the coming “red wave” and what Republicans would do should they reclaim the majority.

But he is often asked whether Mr. Trump intends to run for president.

Mr. McCarthy has told donors that Mr. Trump has not yet made up his mind and that he has advised the former president to see whether President Biden runs for re-election. Mr. McCarthy also often mentions former House members who he said could make for serious presidential contenders, including Senator Tim Scott of South Carolina and Gov. Ron DeSantis of Florida.

On Capitol Hill, Mr. McCarthy’s basic problem comes down to math. Leadership positions in the House can be secured with a majority vote from the members of each party. But the speaker is a constitutional official elected by the whole House and therefore must win a majority — at least 218 votes.

In 2015, after the most conservative House members drove the speaker, John A. Boehner, into retirement, Mr. McCarthy, then the No. 2 Republican, was the heir apparent — and he blew it. His biggest public offense was a television appearance in which he blurted out that the House had created a special committee to investigate the attack on a U.S. diplomatic compound in Benghazi, Libya, expressly to diminish Hillary Clinton’s approval ratings.

“I said multiple times at the time, we need a speaker who can speak,” recalled former Representative Jason Chaffetz, who challenged Mr. McCarthy for the speakership after the gaffe.

Ultimately, Republicans recruited Paul D. Ryan, the Ways and Means Committee chairman and former vice-presidential nominee, for the job.

Republicans who were around then believe Mr. McCarthy has learned his lesson.

“He’s come of age professionally on the math of 218,” said Eric Cantor, a former House majority leader who lost re-election to a primary challenger from the party’s right flank. “He has been schooled in that for many years now.”

Republican leaders are predicting an overwhelming sweep in November’s midterm elections that would give Mr. McCarthy a majority large enough to allow him to shed a few votes and still win, but others in the party are not so sure. The redistricting process has allowed both parties to shore up their incumbents, leaving only a few dozen truly competitive districts. Republicans are still favored to win the majority, but the margin could be slim.

Brendan Buck, a former adviser to two House speakers, Mr. Ryan and Mr. Boehner, said Mr. McCarthy would likely be meticulously shoring up his position.

“He has a system in place that is on top of every member, knowing where they are, how firm their support is for him, and they are working on the members where it’s not strong enough,” he said. “This is not something you just hope works out.”

It appears that Mr. McCarthy has deflected a potential challenge from Representative Jim Jordan of Ohio, who remains closer to Mr. Trump, by successfully pushing for him to become the top Republican on the powerful Judiciary Committee. Mr. McCarthy shows up at meetings of the House Freedom Caucus, the far-right group that is most closely aligned with the former president.

Past tensions with Representative Steve Scalise of Louisiana, the Republican whip and a potential challenger, have for the most part been defused.

“I didn’t know him at all before, and I didn’t take it personally,” Representative Ronny Jackson of Texas, Mr. Trump’s former White House physician, said of Mr. McCarthy, who did not support him in his 2020 primary. “I think he’s earned an opportunity to lead the conference.”

More moderate members also expressed confidence in him.

“Leader McCarthy is very astute, sharp and savvy,” said Representative Peter Meijer of Michigan, another of the 10 Republicans who voted to impeach Mr. Trump. “He has been able to not focus on the differences but find where we can come together on policy choices.”

Yet some Republicans say it can be difficult to discern what principles guide Mr. McCarthy. This month, under intense pressure from the U.S. Chamber of Commerce, he and his leadership team recommended that Republicans vote against a bill abolishing mandatory arbitration in sexual abuse cases, circulating emails noting that Mr. Jordan, the ranking member of the committee that considered it, was opposed.

But when it came to a vote, Mr. McCarthy hung back on the House floor, waiting to register a position until he saw that the bill was passing with overwhelming bipartisan support. At that point, he voted “yes,” leaving some Republicans surprised that he had broken with his own party line.