Cheney to keep fighting Trump after loss, but unclear if she can wield influence over GOP or Dems: experts

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Liz Cheney loses Wyoming primary, slams GOP as ‘cult of personality’

"Special Report" host Bret Baier analyzes Wyoming, Alaska primaries and discusses the effects of the Inflation Reduction Act.

Rep. Liz Cheney has no plans to exit politics after her primary loss, but it's not clear she'll be able to wield the influence she wants in her declared mission to prevent former President Donald Trump from retaking power.

Cheney, R-Wyo., announced Wednesday, the day after her primary defeat, that she will start a new group aimed at opposing Trump. And a Federal Election Commission filing indicated that group will be a leadership PAC called "The Great Task," which is intended to be a channel for fundraising and other political activity for Cheney, as speculation continues about whether she'll run for president in 2024.

"You're looking at a margin of 29 [%] to 66 [%] she lost by. That's just a significant margin. And I think one of the challenges that exist clearly for her is her ability, rethinking her ability to effectively engage the electorate," GOP pollster David Winston of The Winston Group told Fox News Digital on Wednesday. "When you're in your home state and you only get 29% of the primary vote, that's got to be troubling."

Rep. Liz Cheney, R-Wyo., shown, lost her primary race to Trump-backed opponent Harriet Hageman on Tuesday. (AP Photo/J. Scott Applewhite)

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"In coming weeks, Liz will be launching an organization to educate the American people about the ongoing threat to our Republic, and to mobilize a unified effort to oppose any Donald Trump campaign for president," Cheney spokesman Jeremy Adler told Fox News Digital about her next steps.

Leadership PACs like this are common for potential presidential candidates. And Cheney herself is not tamping down on that speculation.

"This primary race is now over, but now the real work begins," she said in remarks conceding defeat to primary challenger Harriet Hageman on Tuesday.

But the Republican Party is still extremely aligned with Trump, as Tuesday's primary results indicate. And Cheney's voting record is closer to Trump and farther from Biden than her successor as House GOP conference chair, Rep. Elise Stefanik, R-N.Y., according to FiveThirtyEight.

Former President Donald Trump endorsed Harriet Hageman, who beat Rep. Liz Cheney, R-Wyo., in a primary. (AP Photo/Seth Wenig)

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Those factors, political experts told Fox News Digital, make it unclear whether there's a lane or constituency for her political message.

"It's a puzzle, because if she runs within the Republican Party, it's just a kamikaze mission and likely the plane will miss," University of Virginia Center for Politics director Larry Sabato said. "If she runs as an independent, she risks taking some critical percentages from the Democratic nominee" to the potential benefit of Trump.

Cook Political Report founder Charlie Cook disagreed on what the implications of a Cheney independent run would be.

"I don't think a lot of Democrats would vote for her if a Democrat were an option," Cook said. "The reason that a lot of them are interested in her is because of her opposition to Trump. But if you had a Democratic opponent, why would many people do that given that she is more conservative than Donald Trump is?"

Rep. Liz Cheney, R-Wyo., will be out of office in January but has not plans to exit politics. (Jim Bourg/Pool via AP, File)

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Cook also said it may not be as far-fetched as it seems for Cheney to successfully run as a Republican in 2024, noting that multiple recent elections have featured surprises in primaries, including 2016 with Trump himself. But Winston, the GOP pollster, said that no matter Cheney's decision, she's unlikely to succeed at stopping Trump without a "serious rethink" to her approach.

"She at this point is around 50,000 votes after having spent $7 million. That's not a very good return at all," Winston said. "To say, 'I'm just going to do the same thing at the national level' — then guess what? You're likely to get the same result."

Winston added: "She just uniquely showed how she can take ample resources and not succeed."

Sabato said he thinks it's unlikely Cheney will "recalibrate much of anything" — an assessment Cook agreed with. "If she ran for president, it would be about Trump," he said.

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Cheney for her part, signaled Tuesday that she does not plan to change her approach of fighting tooth-and-nail against Trump and his allies.

"If we do not condemn conspiracies and the lies, if we do not hold those responsible to account, we will be excusing this conduct and it will become a feature of all elections. And America will never be the same," she said. "We must be very clear about the threat we face and about what is required to defeat it. I have said since Jan. 6th that I will do whatever it takes to ensure Donald Trump is never again anywhere near the Oval Office."

Fox News' Kelly Phares, Paul Steinhauser and Monica Oroz contributed to this report.