Homology group for trivial group action commutes with direct product in second coordinate

Statement

For two groups

Suppose $G$ is a group and $A_1$ and $A_2$ are (possibly isomorphic, possibly non-isomorphic) abelian groups. Let $q$ be a nonnegative integer. Denote by $H_q(G;A_1)$, $H_q(G;A_2)$, and $H_q(G;A_1 \times A_2)$ the homology groups for trivial group action of $G$ on $A_1, A_2$, and the external direct product $A_1 \times A_2$ respectively. Then, there is a canonical isomorphism:

$H_q(G;A_1) \times H_q(G;A_2) \cong H_q(G;A_1 \times A_2)$