Generalized class group actions on oriented elliptic curves with level structure

We study a large family of generalized class groups of imaginary quadratic orders O and prove that they act freely and (essentially) transitively on the set of primitively O-oriented elliptic curves over a field k (assuming this set is non-empty) equipped with appropriate level structure. This extends, in several ways, a recent observation due to Galbraith, Perrin and Voloch for the ray class group. We show that this leads to a reinterpretation of the action of the class group of a suborder O' ⊆ O on the set of $O'$-oriented elliptic curves, discuss several other examples, and briefly comment on the hardness of the corresponding vectorization problems.