  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 $O' \subseteq O$ on the set of $O'$-oriented
elliptic curves, discuss several other examples, and briefly comment on the
hardness of the corresponding vectorization problems.
