Let $h:M\to M$ be a homeomorphism of a compact manifold. Let $p:\tilde M\to M$ be a covering. 1) Is it always possible to lift $h$ to $H:\tilde M\to \tilde M$ so that everything fits into the commutative diagram? 2) Given such a diagram assume additionally that p is a self-covering. Is it true that $H$ is necessarily homotopic to $h$?
1/I think I can see that the answer to the second question is "no". Any additional assumptions that would make it into a "yes"?
2/A reference to the proof of the statement in Ben's second paragraph is needed.