Let $G$ be a finite group. The category of its representations (complex linear, finite dimensional, throughout this whole question) is equivalent to $\mathbb{C}[G]$-modules. V. Drinfeld constructed a quantum double $D(G)$, which is a Hopf algebra, whose category of representations is the categorical center (also called the Drinfeld center) of Rep$(G)$.
The Hopf algebra $D(G)$ can be realized as a crossed module algebra, where the crossed module is given by
$$ 1: G \to G $$
with the conjugation action. By the correspondence between crossed modules and 2-groups [2], one can also realize $D(G)$ as a 2-group, with the set of objects being $G$, and with exactly two morphisms (invertible to each other) between each pair of objects $g$ and $h$. One can show that the Hopf algebra structure of $D(G)$ corresponds to the monoidal structure of the 2-group.
Question
So $D(G)$ is really a 2-group. What I'm interested in is the quantum/Drinfeld double of 2-groups.
(1). Under what kind of representations of 2-groups, does the 2-group have representations correspond to the representations of the Hopf algebra $D(G)$?
(2). Has one constructed a quantum double of 2-groups, whose category of representations in the sense of (1) give rise to the categorical center of Rep$D(G)$?
Possibly related: The character of the 2-representations of a finite group $G$ is a representation of the Drinfeld double $D(G)$. [Theorem on p.7, [1]].
[1] Representation and character theory of finite 2-groups-[Robert Usher]
[2] Notes on 2-groupoids, 2-groups and crossed-modules-[Behrang Noohi]-[arXiv:math--0512106]
