On Friday, Fox also announced that its longtime chief financial officer, Mark Kranz, would retire. His departure was linked to his oversight of the network’s finances during a period when financial settlements were made with women who had complained of harassment, according to two people who requested anonymity to describe internal matters.

Image Jack Abernethy will be in charge of finance, business and legal affairs at Fox News. Credit... Kevork Djansezian/Getty Images for TheWrap

Fox News is a significant source of profit for its parent company, 21st Century Fox, and the Murdochs would prefer smooth operations at the cable channel during a tumultuous election season that has resulted in record ratings. The elder Mr. Murdoch has said he is committed to maintaining Fox’s “distinctive, powerful” voice, curbing the predictions of those who thought that the right-leaning views of Fox’s opinion anchors might soften in the absence of Mr. Ailes.

Mr. Shine, 53, has been with Fox since shortly after the channel debuted in 1996. He is a favored figure among some veteran anchors, including Mr. Hannity, who first recommended him to Mr. Ailes for a job. A Long Island native, Mr. Shine cut his teeth at the network producing Mr. Hannity’s program and working closely with personalities like Bill O’Reilly.

His appointment was widely viewed as a sign of stability at a chaotic time, particularly with newsroom gossip focused on whether top-tier anchors could leave in the wake of Mr. Ailes’s departure.

Since Ms. Carlson went public with her allegations on July 6, a schism has developed within Fox News between Fox News loyalists — some of whom owe their careers to Mr. Ailes — upset at his ouster and others who either did not come forward or were dismayed by those who were defending Mr. Ailes before the investigation was complete.

“I could not be happier with the new management team at Fox News Channel,” the anchor Greta Van Susteren, who also worked closely with Mr. Shine, wrote on Twitter on Friday. “Each is well liked and well respected; Thank you Rupert!”