Let $G$ be a group, and let $M(G)=H^2(G,\mathbb{C}^*)$ be the Schur multiplier of $G$. There is a group $Br(G)$ of complex projective representations of $G$ modulo those that can be lifted to linear representations. Projective representations of degree $n$ are classified by cohomology classes $H^1(G,PGL_n(\mathbb{C}))$. There are maps of pointed sets $$H^1(G,PGL_n(\mathbb{C}))\rightarrow M(G)$$ arising from the exact sequences of (trivial) $G$-modules $$1\rightarrow \mathbb{C}^*\rightarrow GL_n(\mathbb{C})\rightarrow PGL_n(\mathbb{C})\rightarrow 1.$$ Let $Br(G)$ be the subgroup of $M(G)$ generated by the images of these boundary maps for all $n$. (Of course, $Br(G)$ can be defined directly without using group cohomology.)

My question is: when is $Br(G)\rightarrow M(G)$ surjective? This is analogous to the question of Grothendieck on the difference between $Br(X)$ and $H^2(X_{et},\mathbb{G}_m)$ of schemes. If one works with the classifying space of $G$, then everything may be interpreted in terms of topological Azumaya algebras, and the question is precisely the same as Grothendieck's.

I know of three broad cases where this is true. First, when $G$ is a finite group, this follows fairly easily from group cohomology. Roughly speaking, one takes a cocycle representative $\alpha\in Z^2(G,\mathbb{C}^*)$. Then, one uses the cocycle to create a twisted group algebra $\mathbb{C}^{\alpha}[G]$. This algebra is a semi-simple algebra over the complex numbers, so it has various finite-dimensional representations. Any one of them gives a projective representation for $G$ with cohomology class $\alpha$. See for instance Karpilovsky's book *Projective representations of finite groups*.

Second, when $G$ is a group such that $BG$, the classifying space, has the homotopy type of a finite CW-complex, it follows from Serre's theorem that $Br=Br'$ for finite CW-complexes. See Grothendieck's *Groupe de Brauer I*.

Third, when $G$ is a good group in the sense of Serre, then $Br(G)=M(G)$. A group $G$ is good if it has the same cohomology as its finite completion. This result is proven in Schroer's paper *Topological methods for complex analytic Brauer groups*.

So, my real question is: does anyone know any more general results than these? For instance, is $Br(G)=M(G)$ for any finitely presented group?