Following the definition of distinctive filtration, the persistent homology is called index-based if l, p ∈ N, or time-based if l, p ∈ R. Then, we return to the notion of group action for persistent homology: X G is the set of fixed points of the action of G, i.e., X G = {x ∈ X|gx = x∀g ∈ G}. K G denotes the subcomplex of K formed by cell complexes that are pointwise fixed under the action of G. Clearly,for fixed chain complexes, C k (K G ; F) ⊂ C k (K; F) G . Theorem 5.1 (Transfer homomorphism for fixed chain). Let F be a field whose charateristic is either zero or comprime to |Γ|, then there is an isomorphism induced by p,