I'd go with 'f***'s' as the 'sake' belongs to the 'f***'.
Or maybe “sake” refers to rice wine?Or should the apostrophe be after the s - there's usually more than one f*** involved...
It happens. I employed a non resident caretaker for the 11 storey flats I lived in. Very fit young man who came highly recommended, previously worked in a much higher local Council block. Ran up and down stairs all day, impressive. Turned out he was terrified of lifts . We sacked him.
IIRC, I think you said that a few weeks back when the question was posed thenI'd go with 'f***'s' as the 'sake' belongs to the 'f***'.