The Murdochs are still in crisis mode at Fox News. The network was rocked yet again on Friday, this time by a devastating article in New York magazine by Gabriel Sherman, who reported that Fox News paid a $3 million settlement to a booker who said Mr. Ailes carried on an abusive 20-year relationship with her — at times using company resources and personnel to facilitate it. Mr. Ailes denied the charges through his lawyer.  It immediately raised new questions about whether the Murdochs would be forced into a bigger house cleaning of Mr. Ailes’s remaining team of lieutenants at Fox News, where, it seems, sexual harassment payouts were not followed by the recognition that there just might be a problem, let alone by any obvious attempts to aggressively address the corporate culture that facilitated the behavior in the first place.  Mr. Ailes’s departure from the network leaves a huge vacuum in its own right. He drove it to pursue the stories that helped define the strife of the Clinton, Bush (No. 43) and Obama eras. He made it a TV headquarters for the patriotism-infused Bush war marketing effort; the false accusations that Mr. Obama was a “socialist” of dubious citizenship, and, most recently, the Trump movement.  Mr. Murdoch abided by it all, even when it conflicted with some of his own views, such as his support for a path to citizenship for certain undocumented immigrants. But as often as not he seemed to be in line with Mr. Ailes, at least based on how he portrayed himself on Twitter — as a “climate change skeptic”; as a fan of Dr. Ben Carson, “a real black president who can properly address the racial divide”; and as one who believes: “Maybe most Moslems peaceful, but until they recognize and destroy their growing jihadist cancer they must be held responsible.”  Though executives who have worked with Lachlan Murdoch say they assume he shares some of his father’s conservatism, they also say he does not readily advertise his views, which is in itself a major departure from the elder Murdoch.  As for James Murdoch, his leanings are in plainer view than his brother’s, and they are decidedly different from his father’s (with an important exception, friends say: free-market fiscal policy).  James Murdoch’s wife, Kathryn, is a trustee of the Environmental Defense Fund and a former director at the Clinton Climate Initiative. The couple started Quadrivium, a foundation that focuses on the “sustainable use of resources” and “scientific understanding.” James has spearheaded initiatives to make the company “carbon neutral.”  In an essay in Time magazine in December, he wrote, “Entrenched and compromised interests spin the fiction that science is more divided than united, and they sow seeds of uncertainty on issues of unquestionable priority: namely, the survival of our species on this planet.”  His views have heartened producers and executives at National Geographic, which 21st Century Fox took greater control of last year.  Fox News’s reporting often tells a much different story. Its hosts don’t hesitate to report that “the science is still in question,” as Heather Childers recently did, or that warnings about climate change are emanating from “people aligned with the political left in the scientific community,” as the host Steve Doocy said in April while promoting a film purporting to debunk climate change.  Even before Mr. Ailes’s ouster, climate activists were hopeful that James Murdoch would force changes to skeptical coverage of climate change at Fox News, as well as its corporate cousin The Wall Street Journal, whose editorial page has great sway with congressional Republicans.  To nudge it along, one group, Partnership for Responsible Growth, has run ads with both outlets reminding Republicans that their leaders used to support market-based solutions to climate change — and calling on The Journal’s editorial page to acknowledge humankind’s role.  David Fenton, a longtime strategist for progressive causes whose agency, Fenton, made the ads, said its main goal was to push a bipartisan solution in the form of a carbon pricing system. The Murdoch outlets have been standing in the way, he said, by creating a “false reality bubble denying science” that scares amenable Republican politicians away from the cause.  “James knows better, and there’s no way that someone like that could be happy with the terribly negative role the Murdoch properties have played in slowing bipartisan action on climate,” Mr. Fenton said.  None of the Murdochs would comment for this column.  But it’s impossible to talk about the intersection of their political and business interests without lingering for a moment on the latter. Even if some of Fox News’s programming drives one or both of the brothers bananas, it is still a major cash generator for 21st Century Fox, with $1.6 billion in operating profit in fiscal 2015, according to Brian Wieser, an analyst at Pivotal Research Group.  Fox News’s raison d’être was to fill a void that conservative-leaning viewers sensed in the mainstream media. That guiding philosophy has provided a steady compass, and a lucrative revenue stream, as its competitors have struggled with their own paths.  That could provide a strong argument against drastic change, especially for the elder Mr. Murdoch, who is by all accounts enjoying his new role running the network and is saying he won’t rush into picking a successor to Mr. Ailes. It seems a near certainty that he will stay through the election.  With Mr. Murdoch at the helm, the network covered Hillary Clinton’s convention in Philadelphia with all the skepticism you’d expect. It didn’t show the speech of Khizr Khan, the Muslim father of a fallen American soldier. Then, there was Bill O’Reilly’s aside that the slaves who built the White House, whom Michelle Obama referred to in her prime-time address, were “well fed” and adequately housed.  That said, Fox News’s audience tends to be older, which is why executives there indicate that the family views its younger prime-time star, Megyn Kelly, as an important part of its future. Her appeal extends to the core Fox News viewer as well as to those with more mainstream news tastes.  Ms. Kelly’s contract, like Mr. O’Reilly’s, comes up for renewal next year. Before Mr. Ailes left, Ms. Kelly seemed certain to leave. Now it’s a given that her decision will depend on how the Murdochs decide to proceed — as will so, so much else.  A version of this article appears in print on August 1, 2016, on page B1 of the New York edition with the headline: With Ailes Out, and Murdoch In, What’s Next at Fox News?. 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Rupert Murdoch as the network’s interim chief after Roger Ailes’s ouster is a smart move, but Mr. Murdoch’s sons may have a hand in the direction of the empire.