Homology group for trivial group action

Definition
Let $$G$$ be a group and $$A$$ be an abelian group.

The homology groups for trivial group action $$\! H_n(G,A)$$, also denoted $$\! H_n(G;A)$$ ($$n = 0,1,2,3,\dots$$) are abelian groups defined in the following equivalent ways.

Definition in terms of classifying space
$$\! H_n(G,A)$$ can be defined as the homology group $$H_n(BG,A)$$, where $$BG$$ is the classifying space of $$G$$ and theohomology group is understood to be in the topological sense (singular homology or cellular homology, or any of the equivalent homology theories satisfying the axioms).

Definition as homology group for an action taken as the trivial action
The homology groups for trivial group action $$H_n(G,A)$$ are defined as the defining ingredient::homology groups $$H_{n,\varphi}(G,A)$$ where $$\varphi:G \to \operatorname{Aut}(A)$$ is the trivial map. In other words, we treat $$A$$ as a $$G$$-module with trivial action of $$G$$ on $$A$$ (i.e., every element of $$G$$ fixes every element of $$A$$. We thus also treat $$A$$ as a trivial $$\mathbb{Z}G$$-module, where $$\mathbb{Z}G$$ is a group ring of $$G$$ over the ring of integers $$\mathbb{Z}$$.