Assume that I have a short exact sequence of finitely presented groups $$1 \longrightarrow K \longrightarrow H \longrightarrow G \longrightarrow 1,$$ where $G$ is finite (but I do not know whether this is relevant for what follows). Applying abelianization, we get an exact sequence $$K_{\mathrm{ab}} \longrightarrow H_{\mathrm{ab}} \to G_{\mathrm{ab}} \longrightarrow 0.$$

I would like to have a "quantitative" measure of the lack of left-exactness for the above sequence. For instance, I would like to know if it is possible to find an explicit sequence $\{L_i\}$ of (finitely presented) groups sitting in a long exact sequence of the form $$\ldots \longrightarrow L_3 \longrightarrow L_2 \longrightarrow L_1 \longrightarrow K_{\mathrm{ab}} \longrightarrow H_{\mathrm{ab}} \to G_{\mathrm{ab}} \longrightarrow 0.$$

By "explicit" I mean (for instance) that it is, at least in principle, possible to find presentations for the $L_i$ once one has presentations for $K$, $H$, $G$.

Since the category of groups is not abelian (or even additive) we cannot perform the usual construction of the derived functors for the abelianization functor. I am aware that some more refined constructions have been presented (cotangent complex, André-Quillen homology, etc, see for instance the comments to this MSE question) but they look very technical and perhaps overkill in the simple case I have in mind.

I am not an expert, but it seems to me that for the case of groups there should be some more down-to-earth construction, possibly related to the usual group (co)homology, but I looked in some standard textbooks and I did not find any. So, let me ask the following

Question.Is it possible to construct groups $L_i$ as above, in some (at least in principle) computable way, providing a sort of "explicit derived functor" for the abelianization functor? If so, what are some references?