Suppose we have a given action $\varphi : A\rightarrow\text{Aut}(N)$ with $A,N$ abelian groups. Is it possible describe the isomorphism classes of extensions $G$ of $A$ by $N$ realizing $\varphi$ such that the map $G\rightarrow A$ is abelianization (ie, such that $N = G'$)?


(Without the requirement $N = G'$ this is classified by the group cohomology $H^2(A,N)$)


Some remarks:


Nontrivial central extensions certainly give examples where classes in $H^2(A,N)$ do not necessarily satisfy the condition that $G' = N$.


I wonder if there is a condition on $\varphi$ which guarantees that every class in $H^2(A,N)$ represented by $G$ satisfies $G' = N$.



Is it possible to classify extensions $G$ of an abelian $A$ by an abelian $N$ such that the map $G\rightarrow A$ is abelianization?
When $A$ acts trivially on $N$, this $H^2(A,N)$ has a canonical map $\Phi$ into the group $\mathrm{Hom}(\Lambda^2A,N)$ ($\Lambda^2A$ being the second exterior power, quotient of $A\otimes\_\mathbf{Z}A$ by the subgroup generated by elements of the form $x\otimes x$ when $x$ ranges over $A$), induced by the commutator map. Given a central extension with cocycle $c$, the resulting extension has $N$ as derived subgroup iff the corresponding element $\Phi(c)\in\mathrm{Hom}(\Lambda^2A,N)$ is surjective.


(Remark: there's a canonical inclusion $\Psi$ of $\mathrm{Hom}(\Lambda^2A,N)$ into $H^2(A,N)$ and $\Phi\circ\Psi=2\mathrm{Id}$.)


In general, let $N/M$ be the co-invariants of the $A$-action on $N$ (i.e., $M$ is generated as a group by the $\varphi(g)h-h$, when $h$ ranges over $N$ and $g$ over $A$). Then after modding out by $M$, every extension as given yields a cocycle in $H^2(A,N/M)$ and hence, taking $\Phi$ an element in $f\in\mathrm{Hom}(\Lambda^2A,N/M)$; then $N$ is the derived subgroup iff $f$ is surjective.


