Consider a finite abelian group $G$ and a subgroup $H\subset G$, and denote by $A=G/H$ the quotient. Then $G$ is an extension of $A$ by $H$ determined by the short exact sequence $$ 1\rightarrow H\rightarrow G\rightarrow A\rightarrow 1 $$ associated with a class in group cohomology $\epsilon \in H^2(B A, H)$. Thus $G$ is determined by $H,A$ and $\epsilon$. Now let $X$ be a simplicial topological space, and $a\in C^k(X,G)$ a $G$-valued k-cochain. Let us use additive notation for abelian groups. I would like to write $a$ uniquely in terms of a $H$-valued and an $A$-valued k-cochains $b\in C^k(X,H)$, $c\in C^k(X,A)$, using the datum $\epsilon \in H^2(BA, H)$.
Moreover I also need to write the differential $\delta a \in C^{k+1}(X,G)$ in terms of those of $b,c$.
I am pretty sure that this can be done but I don't know a clean and explicit way to do it. Could somebody help me?
The homotopy cofiber sequence associated to $H→G→A$ is $$H→G→A→H[1]→G[1]→A[1]→⋯,$$ which induces a homotopy cofiber sequence of simplicial cochains $$\def\cC{{\sf C^*}}\cC(X,H)→\cC(X,G)→\cC(X,A)→\cC(X,H)[1]→\cC(X,G)[1]→\cC(X,A)[1]→⋯.$$
Thus, $\cC(X,G)$ is the homotopy fiber of the map $\cC(X,A)→\cC(X,H)[1]$. The latter map can be written down concretely as the mapping cone of the map $\cC(X,G)→\cC(X,A)$. This expands to yield precisely the desired description in terms of $A$-valued and $H$-valued cochains.