For an algebraic group $G$ over a field $k$, the abelian category $ Rep_k(G)$ of $k$-linear locally finite representations of $G$ can be identified $QCoh(BG_{lis-et})$. Suppose now I replace $G$ with a group stack $ \mathfrak{G}$, i.e. a group object in the $\infty$-category of $1$-Artin stacks. Then I can consider the abelian category $D_{qc}(B\mathfrak{G}_{lis-et})^\heartsuit$. Here I am thinking of $ B\mathfrak{G} $ as $ [ \bullet / \mathfrak{G}]$ which I suppose will be a $2$-Artin stack.
Question: Is there a way to relate $D_{qc}(B\mathfrak{G}_{lis-et})^\heartsuit$ to some category of "representations" of the group stack $\mathfrak{G}$? (Im not even sure what a representation should be maybe something like a functor $ B\mathfrak{G} \rightarrow D^{[0 ,1]}(Vect_k)$?)
