Let $G$ be a group and $A$ be a $G$-module. I use $H^i(G,A)$ for group cohomology, and $H_i(G,A)$ for group homology. It is well known that if $A$ is a trivial module, then $$ H^1(G,A) \cong \operatorname{Hom}_{\mathbb{Z}}(G^{\operatorname{ab}}, A) $$ Viewing $\mathbb{Z}$ as a trivial $G$-module, it is also well known that $$ H_1(G,\mathbb{Z}) \cong G^{\operatorname{ab}} \cong \operatorname{Hom}_{\mathbb{Z}}(\mathbb{Z}, G^{\operatorname{ab}}) $$ A natural generalization of this second isomorphism is to guess that if $A$ is any trivial $G$-module, then $$ H_1(G,A) \cong \operatorname{Hom}_{\mathbb{Z}}(A,G^{\operatorname{ab}}) $$ The conjecture is further "supported" by the parallel with the $H^1$ isomorphism.
By mimicking the proof of the $H_1$ isomorphism, I was able to prove my conjecture in the case where $A$ is finitely generated. (I can add this if there is interest.)
My main question is, is this well known? I cannot find any references to this generalization in the sources I know for group cohomology and homology. Secondly, is the general case ($A$ not necessarily finitely generated) true?
The correct statement which is well-known is that $H_1(G,A)$ is naturally isomorphic to $G^{ab}\otimes_\mathbb{Z} A$, for any trivial $G$-module $A$. Indeed, this follows from the universal coefficient theorem: the complex whose homology is $H_*(G,A)$ is just obtained from the complex whose homology is $H_*(G,\mathbb{Z})$ by tensoring with $A$, and so we have natural short exact sequences $$0\to H_n(G,\mathbb{Z})\otimes_\mathbb{Z} A \to H_n(G,A)\to \operatorname{Tor}(H_{n-1}(G,\mathbb{Z}),A)\to 0.$$ When $n=1$, $H_{n-1}(G,\mathbb{Z})\cong\mathbb{Z}$ is torsion-free so we just get an isomorphism $H_1(G,\mathbb{Z})\otimes_\mathbb{Z} A \to H_1(G,A)$.
(Note that your proposed generalization doesn't look very natural at all: $H_1(G,A)$ is covariant in $A$ while $\operatorname{Hom}_\mathbb{Z}(A,G^{ab})$ is contravariant in $A$. It turns out they actually are isomorphic if $G$ is finite and $A$ is finitely generated, but the isomorphism is not natural and it is not true more generally.)