Generalized twisted quantum doubles of a finite group and rational orbifolds

In previous work the authors introduced a new class of modular quasi-Hopf algebras $D^{\omega}(G, A)$ associated to a finite group $G$, a central subgroup $A$, and a $3$-cocycle $\omega\in Z^3(G, C^x)$. In the present paper we propose a description of the class of orbifold models of rational vertex operator algebras whose module category is tensor equivalent to $D^{\omega}(G, A)$-mod. The paper includes background on quasi-Hopf algebras and a discussion of some relevant orbifolds.