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 representation of the inertia groupoid of a categorical group. We interpret such a representation as a module over the twisted Drinfeld double $D^\alpha(G)$.
Keywords: categorical groups, representation theory, inertia groupoid, drinfeld double
2010 MSC: 20J99, 20N99
Theory and Applications of Categories, Vol. 31, 2016, No. 21, pp 542-570.