In quantum mechanics, unitary projective representations play a crucial role. To be more general, I want to pose the question in sense of projective representations, then everything would follow as a special case for unitary projective representations.
A projective representation is a group homomorphism $\theta:G \to \text{PGL}(V)$. We thus want to classify projective representations, according to ACuriousMind's post. I am trying to see why $H^2(G,\mathbb{C}^{\times})$ classifies projective representations. I know that $H^2(G,\mathbb{C}^{\times})$ classifies central extensions of $G$ by $\mathbb{C}^{\times}$ up to equivalence. Thus, I am inclined to think that a projective representation is in a one-to-one correspondence with a central extension. How could one write this bijection down explicitly?
Moreover, in order to get the classification, we would have to show that equivalence classes of projective representations (for some suitable notion of equivalence- what does equivalence mean here?) are in one-to-one correspondence with equivalence classes of central extensions (here we know what equivalence means). How would one write down this bijection?
Using the above, we can conclude that equivalence classes of projective representations are in one-to-one correspondence with equivalence classes of central extensions of $G$ by $\mathbb{C}^{\times}$, which in turn is in bijective correspondence with $H^2(G,\mathbb{C}^{\times})$, which would conclude the proof.
Note that I have been told that there doesn't exist a bijection between the two following sets:
\begin{equation} \text{ProjRep}:=\{\theta:G \to \text{PGL}(V)| \theta \; \; \text{is a group homomorphism} \} \; \; \text{and} \; \; H^2(G,\mathbb{C}^{\times}). \end{equation} This is why I am looking for a bijection between $H^2(G,\mathbb{C}^{\times})$ and equivalence classes of $\text{ProjRep}$ under a suitable equivalence relation.
