Continuous orbit equivalence up to equivalence relations

We introduce notions of continuous orbit equivalence and strong (respective, weak) continuous orbit equivalence for automorphism systems of etale equivalence relations, and characterize them in terms of the semi-direct product groupoids, as well as their reduced groupoid $C^*$-algebras with canonical Cartan subalgebras. In particular, we study topological rigidity of expansive automorphism actions on compact (connected) metrizable groups.