Within hours of losing her bid for re-election in Wyoming’s primary this week, congresswoman Liz Cheney converted her campaign finance committee into a political action committee.

“Liz Cheney for Wyoming” became “The Great Task.” A spokesman told Politico that Ms. Cheney “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.”

The move raised more questions than answers, however. How will she do that? Will she snipe at Mr. Trump from the sidelines, or will she wade into the 2024 presidential contest to challenge him if he runs again? Who would vote for her if she did? What exactly is her endgame, many armchairs political strategists are now asking.

Any speculation about Ms. Cheney’s political future on the national stage needs to take into account the fact that only about 170,000 voters participated in the Republican primary in Wyoming and that the state is not representative of the country.

Ms. Cheney lost her re-election bid to Harriet Hageman, a Trump-endorsed challenger. The primary was widely seen as a faceoff between Mr. Trump and Ms. Cheney, who has been among the most vocal opponents of the president within the GOP.

The resounding victory — Ms. Cheney lost by more than 37 points — for Ms. Hageman signals that there is little room in the GOP for opponents of Mr. Trump, or at least for politicians willing to stake their name on opposition to Mr. Trump.

Even with that lopsided a vote, the primary results probably overestimate support for Ms. Cheney within the GOP. Wyoming allows for same day party registration, meaning that thousands of Democrats and independents registered as Republicans to vote that day.

According to the Wyoming Secretary of State, the Democratic party saw a 10,000 voter decline in participation and the Republican party saw a 19,000 voter surge the same day. While it’s impossible to say exactly how same-day registrants split for Ms. Cheney, pre-election polling suggests that they heavily favored her. 

Looking ahead to potential presidential prospects, it’s clear that Ms. Cheney would have to pull out some extraordinary political maneuvering to stand a chance at victory. If that’s her endgame, that is.

That’s because most Republicans prefer a different candidate, and most Democrats prefer a Democratic candidate, according to a political scientist at Hamilton College, Philip Klinker. 

“Where is the constituency for this?” he asks, adding that voters  “have to deal with this problem that only one person can be president. Do you throw your vote away?”

Mr. Kilnker argues that Ms. Cheney understands that her chance of actually winning a race to the White House approaches zero. That is probably not what’s motivating her, though, he said.

“Cheney is not motivated by ambition, because if she was motivated by ambition she would have kept her mouth shut,” he says. “I think she genuinely sees Trump as an acute threat to American democracy.”

That leaves her with a couple options in whatever campaign she mounts to keep Mr. Trump out of the White House. She needs to decide whether to run as an independent or as a Republican.

A political scientist at the University of Georgia, Charles Bullock, believes that Ms. Cheney might wage a campaign not to win but to make life as miserable as possible for Mr. Trump.

He describes a situation where Ms. Cheney is “simply out there trying to destroy Trump, whereas others are trying to get the nomination — almost like an Old Testament prophet warning of the dire consequences should Trump become president.”

Ms. Cheney herself has signaled that she might be up for the task. Rebranding her campaign committee as a PAC allows her to continue raising money and to support candidates or causes.

There is more skepticism about the notion of her running a campaign as a strategic spoiler, trying to siphon votes from Trump. 

An associate editor at Sabato’s Crystal Ball, John Coleman, explains: “Her map — if you look at where she did well in Wyoming, she did well in the few areas that President Biden carried, which tells me that there’s more Biden voters that support her than Trump voters.”

Before she embarks down that path, Mr. Klinker says Ms. Cheney must ask herself whether she would be more likely to draw voters away from Mr. Trump or from Mr. Biden?”

The emerging consensus, based on currently available data from the Wyoming primary and polling on potential presidential primary candidates, suggests that she would likely draw more supporters from Mr. Biden.

A follow up question to this is: could Ms. Cheney runs a strategic spoiler campaign in which she is only on the ballot in certain states? In doing so, she might be able to split the vote in some red states, possibly allowing a Democratic victory.

The best example of a state like this is Utah — a conservative state where Mr. Trump is not especially popular. Senate hopeful Evan McMullin ran such an anti-Trump campaign in 2016, but didn’t even manage to win the state.

The issue, as Mr. Coleman sees it, is that there are not many states like Utah that are both conservative and have any significant anti-Trump constituencies.

Though he is doubtful of such a state-by-state spoiler campaign, Mr. Coleman does recognize Ms. Cheney’s conviction in her task of keeping Mr. Trump out of the White House.

“If you watched right before the primary it’s very clear she wants to run,” he says. “I don’t think we’ve seen the last of Cheney.”