Let ${sigma}(X,x_0,G)$ be the fundamental group of a transformation group (X,G). Let $R({varphi},{psi})$) be the generalized Reidemeister number for an endomorphism $({varphi},{psi}):(X,G){ ightarrow}(X,G)$. In this paper, our main results are as follows ; we prove some sufficient conditions for $R({varphi},{psi})$ to be the cardinality of $Coker(1-({varphi},{psi})_{ar{sigma}})$, where 1 is the identity isomorphism and $({varphi},{psi})_{ar{sigma}}$ is the endomorphism of ${ar{sigma}}(X,x_0,G)$, the quotient group of ${sigma}(X,x_0,G)$ by the commutator subgroup $C({sigma}(X,x_0,G))$, induced by (${varphi},{psi}$). In particular, we prove $R({varphi},{psi})={mid}Coker(1-({varphi},{psi})_{ar{sigma}}){mid}$, provided that (${varphi},{psi}$) is eventually commutative.