Representation and character theory of finite categorical groups

We study the gerbal representations of a finite group $G$ or, equivalently, module categories over Ostrik's category $Vec_G^\alpha$ for a 3-cocycle $\alpha$. We adapt Bartlett's string diagram formalism to this situation to prove that the categorical character of a gerbal representation is a module over the twisted Drinfeld double $D^\alpha(G)$. We interpret this twisted Drinfeld double in terms of the inertia groupoid of a categorical group.