Persistent homology, an extension of algebraic topology, has been developed mainly in the last few decades by Frosini , Edelsbrunner , and others. It is built on a sequence of subspaces or subsets that correspond to homology groups with transfer homomorphisms, and it forms a bridge to other content areas such as measure theory, Riemannian geometry, and computational sciences. As such, it is a powerful mathematical tool for shape comparison. Due to its distinctness, persistent homology has been widely applied for practical use such as shape analysis, image sampling in medical images , and even geometrical and structural analysis for condensed matters, phase transistions, molecular structure . Persistent homology inspects topological spaces and homology groups through functions and derived persistence diagrams. Consequently, in many practical scenarios, it may be appropriate to compare filtration functions that are defined on a topological space with certain groups of transformations . Previous research has utilized persistent homology to capture higher-order structural features as descriptors for data analysis and potential machine learning . These topological invariants, including circles, loops, holes, and cavities, cannot be described by traditional graph and network models. Motivated by its significant applications, we present in this article where persistent homology exposed to spaces with certain symmetry and induced by G-invariant covering maps, like in crystalline phases or clusters in a shape of polyhedron. Specifically speaking, lattice structures are of highly ordered configuration of particles that aggregates according to certain symmetry rules. The physical and chemical properties of the structure are determined by these rules. Since persistent homology incorporates geometric information into topological invariants, it can provide a quantitative description of crystal lattice structures. In conclusion, the significance of this theoretical application may lie in the analysis of chemical or geometrical structures found in certain condensed phases.