Which homeomorphisms $h$ of $S^n\times S^1$ are isotopic to a homeomorphism $g$ of $S^n\times S^1$ such that $g(S^n\times x)=S^n\times x$ for some $x\in S^1$? In particular, does it suffice to assume that $h$ acts trivially on the fundamental group? I would be satisfied with an answer for $n>3$.