I believe that all the Republicans in both houses voted for the repeal that was sent to Obama. So all of them were fakers, not just the Freedom Caucus?
The trickery of such votes became apparent to me during the various votes over Planned Parenthood funding through the decades.
A very rich man now deceased in my area supported the local GOP members of Congress in the past, but he was also a very devoted supporter of Planned Parenthood.
The way it worked was....
Bill Clinton is POTUS.... GOP member votes for cutting PP funding knowing Clinton will veto
George W. Bush is POTUS.... GOP member votes to oppose cutting PP funding knowing Bush will sign it
Barack Obama is POTUS....GOP member votes for cutting PP funding knowing Obama will veto
They all knew Obama would veto repeal of Obamacare it but it made them all look good on Obamacare in time for the election.