If it might hurt their feelings, I don’t say anything. There is something I am not involved with there, and probably shouldn’t be touching if I am not the cause of the problem.
I’d feel miserable telling them the truth, especially if it was my truth, and they already convinced themselves I was someone else. It seems to happen all the time. I don’t like upsetting people, but setting things straight only to make things worse is not a good way to make friends (or keep them).
This is why I try to be as honest as possible from the beginning.
Anything they assume from there is their fault.