Following up on his false claim that at least 3 million illegal immigrants voted in November's election, President Donald Trump was all set last Thursday to sign an executive order initiating a federal investigation into voter fraud.

But the order never came. A spokesman said Trump got stuck in meetings that ran long.

Since then, the White House has moved on to other issues, like banning travel from seven majority Muslim nations and threatening to defund sanctuary cities, without rescheduling the signing. An aide to Trump told NBC News on Friday that there would be no voter inquiry any time soon, although Trump seemed to contradict that in an interview that aired Sunday afternoon.

"I'm going to set up a commission to be headed by Mike Pence, and we're going to get to the bottom of it," Trump told Fox News' Bill O'Reilly, without offering specifics.

One possible reason for the delay: the lukewarm response from within Trump's party. Rep. Jason Chaffetz of Utah, chairman of the House oversight committee, has said he wants no part of the probe. Ohio's Republican secretary of state, Jon Husted, has dismissed the need for it. And the bipartisan umbrella group for state election officials has said it knows of no evidence to support Trump's claims.

Numerous studies have shown conclusively that, although there are rare cases of illegal voting, systemic voter fraud simply doesn't exist.

The White House may simply be waiting until Sen. Jeff Sessions, R-Alabama, is confirmed as attorney general, since he'd likely oversee the probe.

Voting rights advocates, meanwhile, have expressed concern that a seemingly unwarranted investigation would simply lay the groundwork to enact strict new voting laws and policies that end up keeping large numbers of would-be voters from the polls.

Trump expressed staunch support for such laws on the campaign trail and stoked fears of a "rigged" election in the last weeks of his campaign when it was not clear that he would prevail.

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In a tweet on Oct. 17, he declared that there's "large-scale voter fraud happening on and before election day," an unsubstantiated claim he would repeat after winning the election (though not the popular vote) on Twitter and again months later as president.

In January, he used his first official meeting with congressional leaders at the White House to falsely claim that 3 million to 5 million "illegals" voted, costing him the popular vote. Trump, who lost the popular vote by nearly 3 million votes to Democrat Hillary Clinton, has not offered any proof.

Related: Who Is Gregg Phillips, the Man Trump Name-Checked to Prove Voter Fraud?

One scenario, voting rights advocates say, would be an effort to pass a federal law requiring documentary proof of citizenship for anyone registering to vote — effectively a nationwide voter ID requirement for voter registration that they say would make it all but impossible to conduct community registration drives.

Kansas Secretary of State Kris Kobach, the architect of such a law in his state and a close Trump ally, said last week that he's advised Trump to investigate fraud.

The Justice Department could also try to force purges of voter rolls, using a provision of federal voting law requiring states to keep their rolls up to date. In the past, such purges have often removed eligible voters.

"I expect the Trump Civil Rights Division to bring suits under that provision in places with large minority, youth or other Democratic-leaning populations," Samuel Bagenstos, a former top civil rights official during the Obama administration, wrote recently. "The goal of these lawsuits will be to trim the voting rolls."

And it looks highly likely that under Sessions, the Justice Department will be far less aggressive about bringing voting rights cases — sending a signal to states that they can pass new restrictions without the federal government's standing in the way.

Already, the department has signaled that it may change its position on the challenge to Texas' voter ID law, one of the major voting rights cases brought by President Barack Obama's Justice Department. And during his confirmation, Sessions claimed he wasn't familiar with a federal appeals court ruling in August that drew widespread attention, finding that North Carolina's sweeping voting law "targeted African-Americans with almost surgical precision."

In other words, the Trump administration is likely to favor a more restrictive approach to voting. And despite the lack of systemic fraud, it likely will be able to seize on the results of an investigation to justify that approach.

For instance, it's well known that state voter rolls are riddled with errors, and numerous people — including Trump's daughter, son-in-law, top adviser, and Treasury Secretary nominee — are registered to vote in more than one state. That's because people often move from one state to another without canceling their registrations.

Most agree that cleaning up the rolls would be a good idea, but there's no evidence that not doing so has led to fraud. Still, it could allow the Trump administration to muddy the waters by claiming that the probe exposed at least the potential for fraud.

Indeed, White House Press Secretary Sean Spicer has sought to shift the issue away from whether illegal voting had actually occurred.

"You've got folks on rolls that have been deceased or have moved or are registered into counties," Spicer told reporters when the executive order still seemed imminent. "This isn't just about the 2016 election. This is about the integrity of our voting system."

Trump seemed to echo that approach at a Republican retreat more than a week ago, although his exact meaning was unclear. "We also need to keep the ballot box safe from illegal voting," he said. "Take a look at what is registering, folks."

Pence, who Trump says will lead the investigation, pledged a "full evaluation" at the same retreat in recordings leaked to several news outlets, including The Washington Post and The Guardian.

The administration, Pence told Republicans, will "initiate a full evaluation of voting rolls in the country and the overall integrity of our voting system in the wake of this past election."