Action on group cohomology

Question

Let $A$ be an abelian group with an action of $S_3$. Then one has the spectral sequence $H^i (\mathbb{Z}/2, H^j (\mathbb{Z}/3, A))$ converging to $H^{i+j} (S_3, A)$. I want to understand the action of $\mathbb{Z}/2$ on $H^1 (\mathbb{Z}/3, A)$. My question is the following: one can identify $H^1(\mathbb{Z}/3, A)$ with the group $Hom_{tw} (\mathbb{Z}/3, A)$ of twisted homomorphisms. Note that $\mathbb{Z}/2$ acts on both sides of this $Hom$ group and so it acts in the whole $Hom$ group. Is this the correct action? What bothers me is that this identification comes from the standard $\mathbb{Z} [\mathbb{Z}/3]$- resolution of the trivial module $\mathbb{Z}$ and this resolution is not $\mathbb{Z}/2$-equivariant. Thus, a priori it is not clear that this is "the correct" action.