The group $H^2(G,\mathbb{C}^\times)$ plays a rôle in orbifold conformal field theory and is usually known as the *discrete torsion* group. In fact, in this context one actually needs the explicit cocycle and for the case of a finite ~~simple~~ abelian group it is very easy to compute explicitly.

Let $\varepsilon: G \times G \to \mathbb{C}^\times$ be the cocycle. Without loss of generality one can normalise it so that
$$\varepsilon(0,g)=\varepsilon(g,0) = 1$$
for all $g \in G$. With this normalisation the cocycle conditions become, in addition, the following:
$$\varepsilon(g,g)=1 \quad \varepsilon(g,g')= \varepsilon(g',g)^{-1}$$
and
$$\varepsilon(g_1+g_2,g) = \varepsilon(g_1,g)\varepsilon(g_2,g)$$
from where it follows that if $G$ has order $N$, then for all $g,g' \in G$,
$$\varepsilon(g,g')^N = 1$$

Let $G = \mathbb{Z}/N_1 \times \cdots \times \mathbb{Z}/N_k$ be a finite ~~simple~~ abelian group and let $\alpha_i$ be a generator of $\mathbb{Z}/N_i$, so that we can write any element of $G$ as a sum $\sum_i n_i \alpha_i$ where $n_i = 0,1,\ldots,N_i-1$.

Then one finds that all cocycles are given in terms of $B_{ij} = -B_{ji}$ taking the possible values $0,1,\ldots,\mathrm{gcd}(N_i,N_j)-1$, by the formula
$$\varepsilon(\sum_i n_i\alpha_i,\sum_j m_j\alpha_j) = \exp 2\pi\sqrt{-1}\sum_{i,j} \frac{B_{ij} n_im_j}{\mathrm{gcd}(N_i,N_j)}$$

It is the bilinear $B_{ij}/\mathrm{gcd}(N_i,N_j)$ which is called the discrete torsion. It should be emphasised that torsion here is by analogy with the torsion of a connection in differential geometry and not with torsion as in group theory.

If you google "discrete torsion" and "orbifold" you might find suitable references, just like this paper of Vafa and Witten.