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 <del>simple</del> 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 <del>simple</del> 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][1].


  [1]: http://arxiv.org/abs/hep-th/9409188