Consider a finite perfect group $G$ and a $G$-module, $U\left(1\right)$, on which $G$ acts trivially. Here $U\left(1\right)$ is the set of $1 \times 1$ unitary matrices over $\mathbb{C}$.
Are there any examples in which the second cohomology group $H_{2}\left(G,U\left(1\right)\right)$ is non-trivial ? If the answer is positive, is there an explicit expression for such a non-trivial cocycle ?