Let (X, G) be a transformation group and ${sigma}(X,x_0,G)$ the fundamental group of (X, G). In this paper, we prove that the Reidemeister number $R(f_G)$ for an endomorphism $f_G:(X,G){ ightarrow}(X,G)$ is a homotopy invariant. In particular, when any self-map $f:X{ ightarrow}X$ is homotopic to the identity map, we give some calculation of the lower bound of $R(f_G)$. Finally, we discuss commutativity and product formula for the Reidemeister number $R(f_G)$.