$H^n(G,U(1))= H^{n+1}(G,\mathbb{Z})$ only correct for finite group $G$? Universal coefficient theorem

Question

Sorry, I'm not studying mathematics. Maybe my question is pretty trivial. When I meet projective representation in physics, I will use the math tools of group cohomology which I'm not familiar with.

I mainly have following two questions:

1. How to prove $H^n(G,U(1))=H^{n+1}(G,\mathbb{Z})$? Is it only correct for finite group $G$? What's about $G$ is infinite discrete group, like $\mathbb{Z}\times \mathbb{Z}$? Is there general way to relate $H^n(G,U(1))$ with infinite discrete group $G$ to cohomology with coefficient $\mathbb{Z}$ or $\mathbb{Z}_k$. Because I only know how to compute group cohomology with coefficient $\mathbb{Z}$ or $\mathbb{Z}_k$.

2. Is $H^n(G,\mathbb{C}\backslash \{0\})$ same as $H^n(G,U(1))$?

Thanks for your help.