Topologies induced by group actions

We introduce some canonical topologies induced by actions of topological groups on groups and rings. For H being a group [or a ring] and G a topological group acting on H as automorphisms, we describe the finest group [ring] topology on H under which the action of G on H is continuous. We also study the introduced topologies in the context of Polish structures. In particular, we prove that there may be no Hausdorff topology on a group H under which a given action of a Polish group on H is continuous.