Let $G$ be a finite group and $D(G)$ its quantum double. As in my previous question, a typical irreducible representation (finite dimensional over $\mathbb{C}$) is labeled by $(\theta,\pi)$, where $\theta$ is a conjugacy class of $G$ and $\pi$ an irreducible representation of the centralizer of $\theta$.

By reconstruction theorems (cf. Etingof et al. *Tensor Categories*), the category $\operatorname{Rep}D(G)$ is naturally isomorphic to the categorical center of $\operatorname{Rep}(G)$, whose typical objects are in the form $(X,\gamma)$, where $X$ is an object of $\operatorname{Rep}(G)$ and $\gamma$ a half-braiding.

### Questions

Is there a known translation between both descriptions under the natural isomorphism $\operatorname{Rep}D(G) \simeq Z\operatorname{Rep}G$?

More generally, replacing $\mathbb{C}[G]$ by any finite dimensional Hopf algebra $H$, a typical representation of $H$ is a Drinfeld-Yetter module, i.e. a $H$-module with suitable comodule structure. In this case, is there a known translation from the D-Y module description to the center side?

My impression is that $\operatorname{Rep}D(H)$ is wildly unknown for most finite dimensional Hopf algebras $H$. Is this impression correct? Is there at least a criterion for simplicity?