Let $G$ be a group and let $A$ be a $G$-module. Suppose we're given a short exact sequence:
$$1 \to A \to X \to G \to 1$$
It is well known that there exists a bijection between $H^2(G, A)$ and equivalence classes of $X$. Also, the $\mathbb{Z}$-module structure of $A$ makes $H(G,A) = \bigoplus H^i(G,A) $ into a $H(G,\mathbb{Z})$-algebra. I wonder if there is any particular interpretation of elements of $H^2(G,A)$ as the cup-product of elements in $H^1(G,A)$ and $H^1(G,\mathbb{Z})$ - even for particular $G$ and $A$. What I look for is to produce a simple, small and explicit interpretation of cup products in low-dimensions, and the interpretations of the first and second cohomology groups may be the right ground for it.
Interesting examples I guess would happen when $\cup\colon H^1(G, A)\times H^1(G,\mathbb{Z}) \to H^2(G,A)$ is surjective, or when $H^2(G,A)$ is known to be trivial.