I just wanted to report this odd circumstance I found that repeats for me.

1. Complete Operation Anchorage DLC and get the Winterized T-51b armor
2. You have the "Gotta shoot them in the head" quest active (Underworld bartender gives it)
3. In your megaton house, upstairs file cabinet next to your bed, place the Winterized T-51b armor in the file cabinet
4. When you remove the Winterized T51b armor from the file cabinet back to your inventory you will hear the DING and get 300xp with the quest being completed

I haven't actually done any part of the quest other than getting it and the targets are still alive.
This the the Steam version GOTY edition with the autoupdate on.

I just stumbled on it randomly (obviously) but I wanted to report it even if at this point it won't be fixed due to the age of the game.