This trick varies from game to game (it tends to work more often with games on the older side), but here is what you can try:
-Install the game on some unrelated computer (one that will not be under scrutiny)
-Patch it up (patching can be difficult with this method)
-Copy over the directory to the computer under scrutiny
What you now have is all the game files on the new computer without any of the associated registry entries. Most game I've tried this with work fine, but I don't try this with the most recent ones.
Now it's just a matter of obfuscating the game directory. There's no way to hide the increase of hard disk space used, so if the game takes up a lot of space then this can be noticeable. If the GF notices the increase, and wants to hunt it down, it is possible if she is tech-savvy enough. However, if the GF does not, then the only danger is that she will stumble across the folder accidentally, or walk in on him playing the game.
It is possible to try and mask the increase of hard disk space used by installing some other program(s) at the same time, and blaming the increase on that program.
So here is a how-to:
-On your secondary computer, install the game, patch it, etc
-Copy the directory over to the primary computer (this is a lot easier if you can get them on the same LAN, or one is a laptop so you can connect them directly
-On the primary computer, create a new subdirectory in an important-looking folder that could believably have a large size. Copy the game to there. Alternatively, depending on the game, it might be possible to use an external hard drive/memory stick
-On the primary computer, install a new coding program/movie program/other program that you can claim takes up a lot of space
-Play only when there is no danger of the GF interrupting