This is an extended-comment, but I think no:
In homology, we know that $H_1$ is abelianization, $H_2$ is Schur multiplier, but $H_{n\ge 3}$ is ???
In cohomology, $H^1(G;A)$ is split extensions, and this fits well with $Hom(G,A)$ in the UCT. As you mention, $H^2(G,A)$ corresponds to group extensions of $G$ by $A$, and this fits well with $Hom(H_2G,A)$ in the UCT. But as we see, this all ties in to knowing $H^n(G;A)$ and $H_n(G;A)$ very well...
If we look at $H^3(G;A)$ we get crossed module extensions $0\to A\to N\to E\to G\to 0$, and these are cumbersome. We have no nice interpretation for $H^n(G;A)$ for $n>3$, except more crazy-looking exact sequences. This is why I don't expect a "nice" map (i.e. "interpretation" of it) in the UCT to arise.