Actions of homeomorphism groups of manifolds admitting a nontrivial finite free action

In this paper, we study the action of ${\text{Homeo}}_0(M)$, the identity component of the group of homeomorphisms of an $n$-dimensional manifold $M$ with an ${\mathbb{F}}_p$-free action, on another manifold $N$ of dimension $n+k<2n$. We prove that if $M$ is not an ${\mathbb{F}}_p$-homology sphere, then $N\cong M\times K$ for a homology manifold $K$ such that the action of ${\text{Homeo}}_0(M)$ on $M$ is standard and on $K$ is trivial. In particular, for $M=S^n$ a sphere, any nontrivial action is a generalization of the “coning-off” construction.