Thanks, Tyron. It seems that your argument could work well. However, wouldn't there be any problems with the part of the sequence that's not involving groups and homomorphisms?

Thanks, Mark. Unfortunately, the statement does not appear in these books. However, the discussion is very interesting.

Thanks for your comment! I agree with you, the right notion is the one of homotopy pullback. If p is a fibration we have a homotopy pullback though. I've just came across this link in wikipedia which says that the Eilenberg-Moore spectral sequence collapses when the base space is simply connected. In this case I guess that everything works well en.wikipedia.org/wiki/Eilenberg%E2%80%93Moore_spectral_seque‌​nce