An action graph is a combinatorial representation of a group acting on a set. Comparing two group actions by an epimorphism of actions induces a covering projection of the respective graphs. This simple observation generalizes and unifies many well-known results in graph theory, with applications ranging from the theory of maps on surfaces and group presentations to theoretical computer science, among others. Reconstruction of action graphs from smaller ones is considered, some results on lifting and projecting the equivariant group of automorphisms are proved, and a special case of the split-extension structure of lifted groups is studied. Action digraphs in connection with group presentations are also discussed.
