Given an exact sequence of groups $$1\rightarrow N\rightarrow G\rightarrow K\rightarrow 1$$ we call $G$ a central extension of $K$ by $N$ if the image of $N$ is contained in the center of $G$. Central extensions of a group $K$ by another group $N$ have been classified and are in bijection with the 2-cocycles from $K$ to $N$.
Now, given an an exact sequence of groups $$1\rightarrow N\rightarrow G\rightarrow K\rightarrow 1$$ let's call $G$ a derived extension of $K$ by $N$ if the image of $N$ contains the derived subgroup of $G$.
Is there already a theory built around these kind of exact sequences? Is there any literature on these "derived extensions"?
I haven't been able to find anything on them, but I may be searching for the incorrect term.
If there isn't already a theory, can we classify the possible derived extensions of an arbitrary abelian group $K$ by another group $N$?
Some trivial examples are $$1\rightarrow N\rightarrow N\rightarrow 1\rightarrow 1\quad\text{ and }\quad 1\rightarrow 1\rightarrow K\rightarrow K\rightarrow 1$$
for arbitrary $N$ and abelian $K$.
Some less trivial examples are $$1\rightarrow A_n\rightarrow S_n\rightarrow Z_2\rightarrow 1\quad\text{ and }\quad 1\rightarrow V_4\rightarrow A_4\rightarrow Z_3\rightarrow 1$$
Another example is $$1\rightarrow A_n\rightarrow A_n\times Z_2\rightarrow Z_2\rightarrow 1$$ for $n\geq 5$. So we see that two derived extensions of a group by another need not be isomorphic since $S_n$ is not a direct product of $Z_2$ and $A_n$ for $n\geq 4$.
More generally, $$1\rightarrow P\rightarrow P\times A\rightarrow A\rightarrow 1$$ is a derived extension for a perfect group $P$ and an abelian group $A$ since the commutator subgroup of a product is a product of the commutator subgroups.
Clearly, given the trivial group and a perfect group $N$ there is only one derived extension of $1$ by $N$; and that is $N$ itself. But what about in general? I'm wondering if there's a similar characterization as in central extensions or (right-)split extensions.
Any help or reference is appreciated. Thanks.