A $G$-complex is a CW-complex $X$ together with an action of $G$ on $X$ which permutes the cells (I think the action should be continuous). This action induces an action on the cellular chain complex $C_*(X)$. My question is: why the differential is a map of G-modules?
i.e.: why $\partial(ge^\alpha_n)=g\partial(e^\alpha_n)$?