There is the notion of "groups with operators" which signifies a group $G$ together with a morphism of sets $X \to \operatorname{End}(G)$.
It is easily observed that the endomorphism monoid may be formed in any category, and therefore it stands to reason that there may also be rings with operators, graphs with operators etc.
I'd like to ask whether such things are actually in use in ring theory, graph theory etc.
EDIT: As a bonus, there are also "morphisms with operators", that is to say if we have morphisms of sets $X \to \operatorname{End}(G)$ and $X \to \operatorname{End}(H)$, a morphism with operators is a morphism of groups $φ: G \to H$ which commutes with the action of $X$, ie. $φ(xg) = xφ(g)$. From this, we may form a category in the evident fashion. Is this construction being used for other things than groups?
EDIT 2: Of course, there are all sorts of group actions. My question is rather about situations where the more general concept is needed.