Frobenius Monoidal Functors of Dijkgraaf-Witten Categories and Rigid Frobenius Algebras

We construct a separable Frobenius monoidal functor from $\mathcal{Z}\left({\mathsf{V}\mathsf{e}\mathsf{c}\mathsf{t}}_H^{\omega {\mathrm{\mid }}_H}\right)$ to $\mathcal{Z}\left({\mathsf{V}\mathsf{e}\mathsf{c}\mathsf{t}}_G^{\omega }\right)$ for any subgroup $H$ of $G$ which preserves braiding and ribbon structure. As an application, we classify rigid Frobenius algebras in $\mathcal{Z}\left({\mathsf{V}\mathsf{e}\mathsf{c}\mathsf{t}}_G^{\omega }\right)$, recovering the classification of étale algebras in these categories by Davydov-Simmons [J. Algebra 471 (2017), 149-175, arXiv:1603.04650] and generalizing their classification to algebraically closed fields of arbitrary characteristic. Categories of local modules over such algebras are modular tensor categories by results of Kirillov-Ostrik [Adv. Math. 171 (2002), 183-227, arXiv:math.QA/0101219] in the semisimple case and Laugwitz-Walton [Comm. Math. Phys., to appear, arXiv:2202.08644] in the general case.