For a fusion category $\mathcal{C}$ the Brauer-Picard group $\text{BrPic}(\mathcal{C})$ is the group of all invertible $\mathcal{C}$-bimodule categories under multplication $\boxtimes_\mathcal{C}$.

Let the fusion category $\mathcal{C}$ be a $G$-extension of the fusion category $\mathcal{D}$, that is:

$\mathcal{C} = \bigoplus_{g \in G} \mathcal{C}_g$

$\mathcal{C}_e = \mathcal{D}$

It is shown in [Etingoff, Nikshych, & Ostrick, *Fusion categories and homotopy theory* (arXiv:0909.3140), Thm. 6.1] that $\mathcal{C}_g$ is an invertible $\mathcal{D}$-bimodule category for each $g \in G$ and that $\mathcal{C}_g \boxtimes_\mathcal{D} \mathcal{C}_h \cong \mathcal{C}_{gh}$ as $\mathcal{D}$-bimodule categories, hence defining a homomorphism:

$G \to \text{BrPic}(\mathcal{D})$

$g \mapsto \mathcal{C}_g$

(It is further proven that $G$-extensions of $\mathcal{D}$ are in one to one correspondence with morphisms of categorical 2-groups $G \to \underline{\underline{\text{BrPic}}}(\mathcal{D})$ in [ditto, Thm 7.7], where the target is the Brauer-Picard *groupoid*)

**Question:** Suppose the $G$-extension $\mathcal{C}$ is group theoretical, i.e. it is categorically Morita dual to a pointed fusion category. What (if anything) can be said about the resulting homomorphism $G \to \text{BrPic}(\mathcal{D})$? Or the resulting morphism of categorical 2-groups? What if we restrict $\mathcal{D}$ to be pointed?