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 ?
Can't happen because of the universal coeff Theorem for Homology $0\to H_2(G,Z) \otimes U(1) \to H_2(G,U(1)) \to \mathrm{Tor}(H_1(G,Z),U(1)) \to0$. Since $G$ is perfect $H_1(G,Z)$ is zero, since $G$ is finite $H_2(G,Z)$ is finite, since $U(1)$ is a divisible group and $H_2(G,Z)$ is finite, $H_2(G,Z) \otimes U(1) = 0$. So $H_2(G,U(1)) = 0$.