Precise relation between group cohomology and projective representations

Question

In this post and in a comment of this post it has been clarified that there is no bijection between the set of projective representations
\begin{equation}
\text{ProjRep}:=\{\theta:G \to \text{PGL}(V)| \theta \; \; \text{is a group homomorphism} \}.
\end{equation}
and the second cohomology group $H^2(G,\mathbb{C}^{\times})$. I am aware that the there is a bijection between $H^2(G,\mathbb{C}^{\times})$ and equivalence classes of $C^{\times}$ central extensions of $G$. However, my main motivation for studying group cohomology and central extensions is that I am interested in projective representations. In which preicse manner can one relate (possibly irreducible) projective representations and $H^2(G,\mathbb{C}^{\times})$?

Answer 1

There is a exact sequence $1\to\mathbb C^\times\to\mathrm{GL}(V)\to\mathrm{PGL}(V)\to 1$ with trivial $G$-action, which gives rise to an exact sequence (of groups/pointed sets) $$H^1(G,\mathbb C^\times)\to H^1(G,\mathrm{GL}(V))\to H^1(G,\mathrm{PGL}(V))\to H^2(G,\mathbb C^\times).$$ Here, $H^1(G,H)=\hom(G,H)$ when the $G$-action on $H$ is trivial. Thus, we obtain a map $\hom(G,\mathrm{PGL}(V))\to H^2(G,\mathbb C^\times)$. Although the exact sequence already tells you what the kernel is, the cokernel is unclear, since "$H^2(G,\mathrm{GL}(V))$" is not well-defined; nonabelian group cohomology is only defined for $H^0$ and $H^1$.