from 2-cocycle to classifying map

Question

Let $A,E,G:\mathrm{Set}_*\to\mathrm{Grp}_*$ be functors from pointed sets to (discrete) groups ($*=1$) together with natural transformations $i:A\to E, \ p: E\to G$ such that for any set $X$ \begin{equation} A(X)\xrightarrow{i}E(X)\xrightarrow{p}G(X) \end{equation} be a central extension which correspond to a specific (functorial) class $[c]\in H^2(G; A)$.

Now we can extend these functors to $\mathrm{sSet}_*$, then our central extension will become a principal fibration, which determined by a (homotopy class of) classifying map $[w]\in [G,\overline W A]$.

I'm interested in the following question:

Is it possible to derive $[w]$ from $[c]$ somehow?

I see only two possible ways here:

1. Suppose I can find a specific 2-cocyle $c_X\in Z^2(G;A)$, which is not functorial, and cook up a specific (set-theoretic) cross-section $s_X: G(X)\to E(X)$ from it. Then for given simplicial set $X$ I can try to make collection $\{s_{X_k}\}_k$ into a pseudo cross-section, i.e. try to see, whenever they can commute with all degeneracies and faces except (possibly) $d_0$. In case I'm interested in each $s_X$ depends on the choice of order on $X$, so problem reduced to somewhat like ordering a simplicial set, which is not possible for a general $X\in \mathrm{sSet}$

2. We can decompose $\overline W A$ as $\prod_n K(\pi_{n-1} A,n)$ (should be careful about naturality of such decomposition though), so problem reduced to computation of cohomology $H^n(G; \pi_{n-1} A)$ of $G$, considered as simplicial set and finding $[w]$ somewhere there. We can use cobar spectral sequence, or resolve each group $G(X_k)$ and use spectral sequence for bisimplicial group - whatever, it's look tough.

So, maybe there is much easier way that I missed? For example, I know very little about Čech cohomology, our classifying map lives in $\tilde H^1(G;A)$. For ordinary group cohomology (for discrete groups $G$ and $A$) we have a nice epimorphism $H^2(G;A)\to \mathrm{Hom}(H_2(G),A)$, can we have something similar for $\tilde H^1(G;A)$ (or $[G,\overline W A]$ in fact) ?

Edit: Looks like the question is to general in this form. What about the case when $G(X)$ is perfect for any $X$ and short exact sequence is a universal central extension ?