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The arrest of three State Police troopers and the investigation of dozens of others for alleged overtime fraud is part of a “conspiracy” that may go back years, Governor Charlie Baker said Friday.

“My own opinion is, given the widespread nature of this, [and] once we started investigating this — remember, we sent 40 names that we considered to be severe violators here to the attorney general’s office for criminal review — I anticipate this conspiracy goes back a long time,” Baker told reporters after an unrelated event at Fenway Park.

Baker’s comments come amid a broadening federal investigation into potential fraud inside the State Police, where prosecutors say three troopers allegedly bilked taxpayers out of tens of thousands of dollars in 2016 alone.

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The three — David W. Wilson, 57, of Charlton, former trooper Paul E. Cesan, 50, of Southwick, and Trooper Gary S. Herman, 45, of Chester — were among the 40 troopers the State Police have linked to the alleged theft of overtime pay and are now part of a separate probe by the attorney general’s office, according to administration officials.

The criminal charges stem from the now-disbanded Troop E, which had patrolled the Massachusetts Turnpike. The scandal is part of a series of revelations that have brought considerable heat from Democrats on Baker, who appoints the department’s colonel.

In May, the governor announced a restructuring within State Police, the addition of GPS tracking to 1,000 cruisers and the elimination of the troubled troop.

Baker said Colonel Kerry A. Gilpin is also still “looking back” into 2015 records to determine whether other fraud existed.

“And if we find some of the violations there, we’ll look back into 2014,” Baker said. “... We’re going to continue to go back into previous years and review those records.”

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Reach Matt Stout at matt.stout@globe.com. Follow him on Twitter @mattpstout