Counterexample, where every projective representation can be lifted to a linear representation, but $H^{2}(G,\mathbb{C}^{\times}) \neq \{e\}$.

Question

I am currently learning about projective representations and I have seen a proof that if $H^2(G,\mathbb{C}^{\times})=\{e\}$, then every projective representation can be lifted to a linear representation. However, I have not found anywhere that this statement would be an iff. I am trying to find a counterexample, for which it holds that every projective representation can be lifted to a linear representation, but $H^2(G,\mathbb{C}^{\times}) \neq \{e\}$. Does anyone have a hint in mind, or a reference?

I know that projective reps. are in 1-1 correnspondence with linear reps of $\mathbb{C}^{\times}$ extensions, but I don't know how this helps in this particular case.

Answer 1

The group $H^2(G,{\mathbb C}^\times)$ is the Schur Multiplier $M(G)$ of $G$, and $G$ has a Schur Cover $H$ with a subgroup $M \le Z(H) \cap [H,H]$ of $H$ with $M \cong M(G)$ and $H/M \cong G$.

If $M(G)$ is nontirivial, then $H$ has an irreducible complex representation $\rho$ with $M \not\le \ker \rho$, and then the projective representation $\bar{\rho}$ of $G$ induced by $\rho$ does not lift to an ordinary representation of $G$.