She had the Flip 3 and the cheap housing was way too big. The new Flip 5/6 come with the adapters to fit on the Hero 3 and up (none for 3, thick for 4, and skinny for 5/6) not sure about the 3.1. There's nothing magic about what BackScatter did with the adapters. They have tabs and a cutout to make installation easier and fit better. Nice features and well made but for what the intended purpose is, not completely necessary. If I was to do it over, plastic shim material and some scissors. If your 4 can fit over the housing lens, then just find some plastic shim material and you're set.
Are you speaking to me? If not, apologies but if so, I think you misunderstand.
I stated that I have both frames and all the adapters. The Flip 4 fits on my housing with the thin adapter. The Flip3.1 does not. Clearly something is different. I provided the actual measurements taken with calipers ten minutes ago. The fact that both frames fit the official housings with adapters does not mean the frames are the same. Close, but not the same. If you look at your frame, they both have an allenhead bolt that unscrews to allow the frame to expand open (almost 2mm) and then tighten it down on the lens. There is enough flex there when on the official case to take up a slight difference in the frame size, but not with the knock-off.