When did the IMF become suspicious of Walker in Mission Impossible: Fallout? To me, that plot twist just came out of nowhere.
It is hinted in the film that Sloan had went directly to Alan Hunley when Walker told Sloan that Ethan Hunt is John Lark with the doctored evidence of the cell phone he retrieved in Paris.
My evidence for this comes when Alan Hunley confronts Hunt that the CIA suspects Hunt is Lark. Walker is there to witness this. Hunt incapacitates Hunley and tells Walker to watch Lane while he goes to get Lark and continue the mission.
As we know from the film. Spoilers ahead, although the question itself is a bit of a spoiler.
Walker reveals that he is Lark to Lane (actually Dunn disguised as Lane).
When Walker realizes this Sloan appears on a video phone screen from Hunley's phone and acknowledges that Hunley was correct that Walker was Lark. It is this interaction where I am deducing that Sloan went to Hunley who must've vouched for Hunt and convinced Sloan with this plan to determine if Walker was truly Lark trying to frame Hunt.