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Assume $f:G\rightarrow H$ is a continuous epimorphism of topological groups and $K\leq H$ is a subgroup of $H$. One can consider the actions of subgroups $K$ and $f^{-1}(K)$ on $H/K$ and $G/f^{-1}(K)$ respectively (here $H/K$ and $G/f^{-1}(K)$ denote right coset spaces - with quotient topology and the actions are given by left multiplications).

Consider the fixed point sets of these actions, $Fix(G/f^{-1}(K),f^{-1}(K))$ and $Fix(H/K,K)$ with the induced subset topologies.

Is it true that $Fix(G/f^{-1}(K),f^{-1}(K))$ is homeomorphic to $Fix(H/K,K)$? If not in egneral, then maybe at least in the case $K$ and $f^{-1}(K)$ are finite?

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    $\begingroup$ Is $f$ a homomorphism? Then the fixed point sets are $N_H(K)/K$ and $N_G(K)N/KN$, where $N=\ker(f)$., And these are isomorphic groups (via $f$). $\endgroup$ – Hempelicious Dec 21 '18 at 2:11
  • $\begingroup$ Thinking about this more, $f$ always induces a continuous bijection between the fixed point sets. If $f$ is a quotient map, then you definitely get a homeomorphism. But it's not always true: let $G$ be $\mathbb{R}$ with the discrete topology, $H$ be $\mathbb{R}$ with the usual topology, $f$ the identity map and $K$ the rationals. $\endgroup$ – Hempelicious Dec 21 '18 at 17:20

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