I have the same problem here. The 3-11/16 is definitely the right size but for some reason the whole itself is about 4" in size. On my door it appears thee are holes for some kind of face plat to keep the lock in place. Picture attached.
In theory, the faceplate is not needed because as soon as you install the handle into the lock it keeps it in place. It should only be an issue if you remove the handle and the lock falls into the door (which happened to us).