Consider an exact sequence of groups \begin{equation} 1\rightarrow H\rightarrow K\rightarrow G \rightarrow1~. \end{equation} Mackey theory enables us to understand representations of $K$ in terms of representations of $H$ and $G$.
Specifically, let $V$ be a unitary irreducible representation (over $\mathbb{C}$) of $H$. The group $K$ acts on the isomorphism type of $V$ via conjugation. Let $S\subset K$ be the stabilizer of $V$ under this action. Then we may ask if $V$ is in fact a representation of $S$. In general, we find that it is only a projective representation with an associated cocycle that can be expressed as a pullback from a class $\eta \in H^{2}(S/H,U(1))$. This $\eta$ is the Mackey obstruction class. Correspondingly, if we take a representation of $K$ which upon decomposition to $G$ includes $V$ we naturally find projective representations of $S/H.$
I am interested to know if there is a version of this construction that captures more refined notions of projective representations, perhaps using group cohomology with more general coefficients.
As a specific example that I would like to capture, consider the sequence \begin{equation} 1\rightarrow \mathbb{Z}_{n}\rightarrow U(1)\rightarrow U(1) \rightarrow1~. \end{equation} Every irreducible representation of $\mathbb{Z}_{n}$ is stabilized by $K=U(1)$ and of course can be extended to a representation of $K$. Thus the Mackey obstruction class vanishes reflecting the fact that $H^{2}(U(1),U(1))$ is trivial.
Nevertheless there is a difference between the representations of the quotient $G$ and the group $K$ in this example. The representations of $K$ that are pulled back from $G$ have charges which are multiples of $n$.
Is there a general version of the Mackey obstruction class that incorporates both this example and the previously discussed projective representation theory?
Does the center of $H$ play a distinguished role? (perhaps there is in general an $\eta \in H^{2}(S/H,Z(H))$??)
Any pointers to references would also be appreciated!
