In this article, we derive and implement persistent homology theory to quotient spaces with G-invariant covering by inducing mappings between spaces and calculating their chain complexes, and futher examine their validity by for discrete group actions: (i) A filtration function F and a G-invariant covering map (quotient) p are commutative, and an induced filtration function of distances on X/Γ by isometries can always be built by isomertries on metrizable spaces. (ii) Normal covering series can be constructed to examine properties relating to symmetry from a topological perspective and covering spaces, especially for finite groups and crystallographic groups: