Suppose $G$ is a compact Hausdorff (not necessarily Polish) group, acting continuously on a Polish (not necessarily compact) space $X$ (i.e. the map $G\times X\to X$ is continuous).
Is it true that there is a compact Polish group $H$ acting on $X$ through which the action of $G$ factors?
This is seems true if $X$ is compact, because the action induces a continuous homomorphism from $G$ into the homeomorphism group of $X$, which is Polish in this case, so we can take the image of $G$ for $H$. I suppose more general conditions which force the homeomorphism group to be a Polish group with compact-open topology would suffice as well, but I have no idea about the general case.