Let $U(1)$ denote the multiplicative group of complex numbers of modulus $1$ and $\mathbb Z_2:=\mathbb Z/(2\mathbb Z)=\left(\{0,1\},+\right)$
I know that the central group extension $$\mathbb Z_2\overset{f}\longrightarrow U(1) \overset{g}\longrightarrow U(1)$$ with $$f:\mathbb Z_2\to U(1):n\mapsto e^{in\pi}$$ and $$g:U(1)\to U(1):z\mapsto z^2$$ is non-split, and a possible cocycle for it is
$$c: U(1)\times U(1)\to \mathbb Z_2: (e^{it_1},e^{it_1})\mapsto \begin{cases} 0 & -\pi<t_1+t_2 \le \pi \\ 1 & \text{otherwise} \end{cases}$$
where $-\pi<t_1,t_2\le\pi$. This means that $H^2(U(1),\mathbb Z_2)$ is not trivial. But what is exactly $H^2(U(1),\mathbb Z_2)$? How can it be calculated?