This is a refinement of my older question A homeomorphism with a prescribed action on the fundamental group - decidable or not?
The problem under considreation is the following. Let $M,N$ be two closed manifolds. For the sake of convenience, we assume that they are smooth and of dimension at least 5. We have an explicitly given isomorphism of fundamental groups $p: \pi_1(M)\to \pi_1(N)$ and want to find out if there is a homeomorphism $\phi: M\to N$ such that $\phi_*=p$. Question: for which groups this problem is decidable?
It is known due to the work of Nabutovsky and Weinberger that it is decidable when the group is trivial (or, by extension, finite). On the other hand, they constructed an example demonstrating that it is undecidable when the word problem for $\pi$ is undecidable. One may wonder if it is the only obstacle, and regardless of the answer it seems an interesting problem to me.
PS In case the answer is not known, I would appreciate expert opinions.
