For reasons arising in algebraic topology, I'm wanting to better understand the relations between two functors from groups to abelian groups, $\mathbb{Z}[\cdot]$ and $\operatorname{ab}$; group ring and abelianization.

Specifically, we can extend the homomorphism $G\to \operatorname{ab}(G)$ to an additive map $\mathbb{Z}[G]\to \operatorname{ab}(G)$, and this map is clearly natural. It seems too vague to ask "what should I do with this next?", so instead the question is

What references are there that consider this natural transformation?

For instance, I'd be immensely pleased if this is a good way to start a free resolution of $\operatorname{ab}$ without forgetting too much about $G$, and to learn what's the natural way to continue... but I'm not picky! You'll understand when I complain that automatic search-engines aren't too helpful in looking for anonymous natural transformations.

Thanks.