First homology group for trivial group action equals tensor product with abelianization

Statement
Suppose $$G$$ is a group and $$A$$ is an abelian group. Denote by $$H_1(G,A)$$ the first homology group for the trivial action of $$G$$ on $$A$$. We then have:

$$H_1(G,A) \cong G^{\operatorname{ab}} \otimes_{\mathbb{Z}} A$$

where $$G^{\operatorname{ab}}$$ is the abelianization of $$G$$, i.e., the quotient group of $$G$$ by its derived subgroup, and $$\otimes_{\mathbb{Z}}$$ denotes the tensor product of abelian groups.

Related facts

 * First cohomology group for trivial group action is naturally isomorphic to group of homomorphisms