Using an A/C Register for Exhaust

I want to vent my closet. I need to get cool air into the closet, use a carbon filter to get rid of the odor, and then exhaust it.
I plan to get cool air in passively, by installing an A/C register on the back wall of the closet that will then open into a foyer. Just cut the sheetrock in the closet and in the hallway, and install the register. No issue there.
I then want to hook up my 6" inline fan with carbon filter in the closet, again no issue.
So the air will be pulled in passively from the foyer into the closet. It will then be sucked into the carbon filter by the inline fan pulling it, and then the inline fan will push the filtered air into the flex duct, to the register box, and out into the room the closet serves.

This is the problem. I cannot figure out how to attach the register box to the inside of the wall. These register boxes are apparently designed to be nailed/screwed into the beams in the attic on the SIDES of the register box, but as I am going through sheetrock I don't have that option. Plus the side of the void between the sheetrock has an electic line powerig an outlet, so I cannot nail it to that stud either.

So how do I attach the register box to the sheetrock?

Bump.....No Ideas? I was thinking this is one of the most creative groups ever, LOL....

drill hole fool with drilll bit for stone.

then drive a screw in there pull it back out and put a fresh one in if you dont have a tap kit

You can take couple of pieces of 1x2 wooden blocks and put them on the back side of the sheet rock run some drywall screws into it then there is something to screw your register into.