Both are senior managers and have a long track record with Fox.
Mr Ailes resigned in July after a series of sexual harassment allegations against him by female news presenters.  He has denied any wrongdoing.
Fox News plays a prominent role in US election coverage.
The channel's more conservative tone is popular among many Republican voters.
Mr Abernethy is chief executive of Fox Television Stations, running the 28 Fox-owned channels, and Mr Shine is senior executive vice-president and close aid to the company's founder Rupert Murdoch .
The men will divide the tasks of running the Fox News Channel and Fox Business Network. Mr Abernethy will focus on finance, advertising sales and distribution, while Mr Shine will manage the programming.
"While this has been a time of great transition, there has never been a greater opportunity for Fox News and Fox Business to better serve and expand their audiences. Together with our worldwide team of dedicated professionals, we are the most influential and powerful brand in television," said Mr Murdoch.
Mr Murdoch has taken over the role of Fox News CEO and executive chairman since Mr Ailes stepped down.
Mr Ailes had run the Fox News Channel since its beginning in 1996.

The Tennis Integrity Unit said Lindahl plotted to throw a match at a minor Australian tournament in 2013.
Two other Australian players, Brandon Walkin and Isaac Frost, were also disciplined over the incident.
It comes days after Australian Open junior champion Oliver Anderson was charged with match-fixing.
Last year, a joint BBC/Buzzfeed investigation alleged the TIU failed to act on suspicions that 16 top-50 ranked players have been involved in match-fixing.
Lindahl, 28, achieved a career-high ranking of 187 in singles before retiring in 2013.
The TIU, which is responsible for policing tennis, said the offence happened at the Australian F6 Tournament in Toowoomba, Queensland.
"He was found guilty of charges of contriving or attempting to contrive the outcome of an event, and failing to co-operate with a TIU investigation," it said in a statement.
The penalty means Lindahl will not be able to resume his international playing career or attend official tennis events for seven years, the TIU said.
He had already been fined A$1,000 by a court in New South Wales last year.
Walkin, 22, was handed a six-month suspension for "passing a corrupt proposal to another party" on behalf of Lindahl.
But the 1,066-ranked player is free to compete after his penalty was suspended for six months, subject to further breaches.
Frost, 28 and ranked 1,515, refused to hand over his mobile phone during the investigation. He will not be further penalised after serving a provisional suspension between October 2013 and September 2014.
Twenty-First Century Fox has appointed Jack Abernethy and Bill Shine as co-presidents of Fox News, replacing Roger Ailes who stepped down last month after a sexual harassment scandal.
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Australian former tennis player Nick Lindahl has been banned for seven years and fined $35,000 (Â£28,000; A$47,700) for match-fixing.