I have a metrizable group $G$ (locally compact, if that's useful). I would like to construct a compact Hausdorff space $X$ such that there is a surjective continuous group homomorphism from $G$ to $H(X)$, the group of autohomeomorphisms of $X$ with the compact-open topology. (Any continuous group homomorphism $G$ to $H(X)$ corresponds to a group action of $G$ on $X$ that preserves the topologies.)
Is there a general recipe to achieve this? I am looking for a relevant reference.