Let $G$ be a finite 2-generated metabelian group, and let $S$ be a schur covering group, so that we have an exact sequence $$1\rightarrow M(G)\rightarrow S\rightarrow G\rightarrow 1$$ where $M(G)$ is the Schur multiplier of $G$, and $M(G)\le S'$ (The $'$ denotes derived subgroup). This implies that $S^{ab}= G^{ab}$, and we have an exact sequence $$1\rightarrow M(G)\rightarrow S'\rightarrow G'\rightarrow 1$$ Thus, since $G$ is metabelian, $G'$ is abelian, so $S'$ is also a metabelian group, and is equal to the exterior square $G\wedge G$ of $G$.
It seems reasonable that we should be able to say something about $(G\wedge G)^{ab},(G\wedge G)'$ in terms of $G',G^{ab}$,...etc, though I'm having some difficulty getting an intuitive understanding of what this exterior square "is", and how to work with it.
Is there a good book that discusses nonabelian exterior products of groups?