  We introduce notions of continuous orbit equivalence and strong (respective,
weak) continuous orbit equivalence for automorphism systems of \'{e}tale
equivalence relations, and characterize them in terms of the semi-direct
product groupoids, as well as their reduced groupoid $C^*$-algebras and the
associated $C^*$-automorphism systems of group actions or coactions on them. In
particular, we study topological rigidity of expansive automorphism actions on
compact (connected) metrizable groups
