I just did Guess Who I Saw Today, and decided to let Lily keep the stealth boy mk II. However, I don't see any difference. I expected her to use the thing in battle or when I was sneaking, but she didn't. Is this a bug or didn't Lily's story end yet? She also hasn't received a perk like some other companions do after their quest.

Lily's quest hasn't ended. She doesn't get a perk when it does, though, just her behavior changes.