Scottish Book Problem 166

This is one of the few problems from the Scottish book that are still open. In slightly modernized form, and correcting the typo (in the book, $f$ and $f_0$ should be switched in the last sentence) the problem is:

Let $M$ be a topological manifold, and let $f\colon M\to\mathbb{R}$ be a continuous function. Let $G^M_f$ denote the subgroup of homeomorphisms $T\colon M\to M$ that satisfy $f\circ T=f$ . Let $N$ be another manifold that is not homeomorphic to $M$ . Does there exist a continuous function $f_0\colon N\to\mathbb{R}$ such that $G^M_f$ is not isomorphic to $G^N_{f_0}$ ?

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