From the adjunction $U:\mathsf{Grp}\leftrightarrows\mathsf{Set}:F$ between the free group functor and the forgetful functor, one gets a monad $F\circ U:\mathsf{Grp}\to\mathsf{Grp}$, which we can write just $F$. Given a group $G$, there is an associated (augmented) simplical group $F^\sharp G$, the bar construction for the monad, which looks like the one you wrote.

If $M$ is a $G$-module, them $M$ is a $F^nG$-module for all $n$, by pulling back along the composition of maps going from $F^nG$ to $G$ in the simplical group. It makes sense to talk about derivations from the $F^nG$ to $M$, and in fact we get a cosimplicial abelian group $\operatorname{Der}(F^\sharp G,M)$. From it we can construct as usual a cochain complex (whose differentials are alternating sums of transposed face maps in the simplicial group $F^\sharp G$)

(I have not completely check the details but) the cohomology of the complex $\operatorname{Der}(F^\sharp G,M)$ should be isomorphic to the cohomology of $G$ with values in $G$ up to a shift and a twist: explicitly, $$H^i(\operatorname{Der}(F^\sharp G,M))\cong\begin{cases}Der(G,M), &\text{if $i=0$;}\\H^{i+1}(G,M), &\text{if $i>0$.}\end{cases}$$
This is the usual group cohomology of $G$, but we dropped the $0$th group, and instead of having derivations modulo the inner ones, we have just derivations. Most of the work needed to check this is of the general nonsense kind.

If you apply a more interesting functor $\mathsf{Grp}\to\mathsf{Ab}$ to $F^\sharp G$, you'll get other things, of course.