Fox News host Sean Hannity, an ardent critic of Robert Mueller’s investigation into whether President Trump’s campaign colluded with Russia to influence the 2016 election, has recently cast aspersions on Mueller’s tenure as a federal prosecutor in Boston decades ago.

Hannity has tried to link Mueller to New England’s most notorious organized crime figure: James “Whitey’’ Bulger. In one segment, Hannity presented a sarcastic chart, dubbed “The Mueller Crime Family?’’ that listed Bulger as a member.

During one opening monologue this week, Hannity said it doesn’t seem to matter how “rogue’’ Mueller is. “It doesn’t matter what he did with Whitey Bulger.’’ He also said that while Mueller was a prosecutor in Boston, there were “four men, wrongfully imprisoned for decades, framed by an FBI informant and notorious gangster Whitey Bulger, all while Mueller’s office looked the other way.’’

“In that case with Whitey Bulger — a $100 million payout and two of the four people died in jail that were put in there and they were innocent,’’ Hannity added.

So what’s Hannity talking about, and is what he said true? Here is what we know about Mueller’s record on those matters during his time in Boston.

What involvement did ­Mueller have with Bulger?

None. Mueller served in the US attorney’s office in Boston from 1982 to 1988, as chief of the criminal division, first assistant US attorney, and as acting US attorney for more than a year. During that time, Bulger ran a sprawling criminal enterprise and got away with murders because he was a longtime FBI informant who corrupted his handlers. The FBI and the New England Organized Crime Strike Force, a prosecutorial unit that worked independently of the US attorney’s office and reported directly to the Justice Department, used Bulger to build cases against the Mafia and gave him a pass on his own crimes. The FBI’s corrupt relationship with Bulger was exposed after he was indicted on federal racketeering charges in 1995 and became a fugitive. He was captured 16 years later.

Were four men framed by an FBI informant and wrongfully imprisoned for years, while two died in prison?

Yes, but that informant was not Bulger. Mob hitman-turned-government witness Joseph “The Animal’’ Barboza testified in a 1968 trial that led to the wrongful convictions of Joseph Salvati, Peter J. Limone, Louis Greco, and Henry Tameleo for the 1965 slaying of a small-time hoodlum named Edward “Teddy’’ Deegan. Tameleo and Greco died in prison. The men proclaimed their innocence, but members of the FBI, the US attorney’s office, and the Suffolk district attorney’s office vigorously lobbied against clemency for them throughout the 1980s and 1990s.

Their case drew scrutiny after details of the FBI’s corrupt relationship with Bulger and his sidekick, Stephen Flemmi, began to emerge in 1998, triggering an investigation into the agency’s mishandling of informants dating to the 1960s. In 2000, a Justice Department task force uncovered secret FBI documents indicating Barboza framed the four men, while protecting one of the real killers — Flemmi’s brother. Limone was freed in 2001, and Salvati had been pardoned in 1997.

Did Mueller know the four men had been wrongly convicted and look the other way?

There’s nothing linking ­Mueller to that case, according to several attorneys for the men, voluminous court records, and a former federal judge who presided over their wrongful imprisonment trial. In 2007, then US District Judge Nancy Gertner found that the FBI deliberately withheld evidence that the four men were innocent and that the bureau helped cover up the injustice for decades. She ordered the government to pay the men and their families $101.7 million.

“Absolutely nothing in the record that I saw suggested Mueller’s involvement in any way in either the initial acts that led to the four men’s imprisonment, or the acts that ended in their continued imprisonment and denying them parole or the coverup,’’ Gertner said Friday.

Was Mueller among the prosecutors who wrote letters to the Massachusetts parole board opposing the release of the four before evidence emerged that they had been framed?

No, according to Gertner and Limone’s attorney, Juliane Balliro, who scoured copies of the parole board records for the four men. There were no letters from Mueller in the files and his signature “never appeared on anything I ever saw or can recall,’’ Balliro said.

Former Massachusetts Parole Board member Michael Albano, who complained of intimidation and retaliation by the FBI after he voted in favor of commutation for Limone in 1983, said Thursday that he’s convinced that at one time he saw a letter from Mueller, written in the 1980s, opposing the release of one of the four men.

A 2011 column by the Globe’s Kevin Cullen has been cited in recent media reports that attempt to link Mueller to the wrongfully imprisoned men. At the time, Cullen said Mueller wrote letters to the parole and pardons board throughout the 1980s opposing clemency for the men. But, in a column Friday, Cullen said he heard that from Albano but did not see any letters from Mueller.

After the FBI was found responsible in 2007 for framing the men, Mueller, then the FBI director, characterized the case as a debacle.

Martin Finucane of the Globe staff contributed to this report. Shelley Murphy can be reached at shelley.murphy@globe.com.