TITLE: $\text{Rep}(D(G))$ as representation category of a vertex operator algebra
QUESTION [17 upvotes]: The category of representations $\text{Rep}(D(G))$ of the quantum double of a finite group is well-known to be a modular tensor category. Can these modular tensor categories also be obtained as representation categories of vertex operator algebras?

REPLY [6 votes]: A lot has happened in the last four years, and we now have lots of positive results.
The current state of knowledge is given in Evans-Gannon, "Reconstruction and Local Extensions for Twisted Group Doubles, and Permutation Orbifolds".  In particular, if $G$ is a finite solvable group, then $D(G)$ (and more generally, any twist $D^\omega(G)$) is the representation category of some vertex operator algebra (in particular, the fixed points of a $G$-action on some holomorphic vertex operator algebra).
For non-solvable groups, the result you want would follow from the conjectured regularity of fixed points (i.e., a suitable generalization of C-Miyamoto).
Oddly enough, it turns out that permutation orbifolds can have non-trivial twists.  This is discussed in the Evans-Gannon paper, and earlier in Johnson-Freyd's "The Moonshine Anomaly".