The abelianization functor from the category of groups to that of abelian groups is right exact in the sense that it takes a short exact sequence $$1 \to K \to G \to H \to 1$$ to a shorter exact sequence $$K_{\mathrm{ab}} \to G_{\mathrm{ab}} \to H_{\mathrm{ab}} \to 0.$$ (See Show that the abelianization functor is right exact.) If the category of groups were an abelian category, then we would be able to define derived functors of abelianization. The category of groups isn't actually abelian, or even additive, but at least exactness still makes sense, so one might hope something would go through anyway.
1. What breaks down?
2. What can be salvaged?
3. Is there some morally allied modification to this naive hope that is actually studied?
The most natural extension of the exact sequence is the Lyndon–Hochschild–Serre spectral sequence, which states that for a $G$-module $A$, we have the following spectral sequence: $$E^2_{pq}=H_p(H,H_q(K,A))\implies H_{p+q}(G,A),$$ which is the Grothendieck spectral sequence for the composition of $G\mathrm{-mod}\to H\mathrm{-mod}:A\mapsto A^K$ and $H\mathrm{-mod}\to \mathrm{Ab}:A\mapsto A^H$ (alternatively, the Leray spectral sequence for the fibration $BK\to BG\to BH$).
In particular, it gives the following 5-term exact sequence: $$H_2(G,A)\to E^2_{20}\to E^2_{01}\to H_1(G,A)\to E^2_{10}\to 0.$$ Applying the exact sequence to the trivial $G$-module $A=\mathbb Z$ gives $$H_2(G,\mathbb Z)\to H_2(H,\mathbb Z)\to H_1(K,\mathbb Z)=K_{ab}\to H_1(G,\mathbb Z)=G_{ab}\to H_1(H,\mathbb Z)=H_{ab}\to 0.$$ This is only a very small portion of the spectral sequence, and much more information can be extracted in general.