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?