We have inverstigated the relationship of filtration functions and persistent homology groups on quotient/orbit spaces and their covering, showing their strong correspondence by constructing homomorphisms. As a chracterization of geometrical shapes, persistence diagrams and barcodes are well defined under group actions and of good stability. These results provide useful insight into analyzing structures or networks of certain symmetry in chemical or material sciences. Futher research may be required for persistent homology of specific coefficient field as Z or on topological spaces of sphere S k etc., from some algebraic, homological and computational perspectives.