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We construct a separable Frobenius monoidal functor from$\mathcal{Z}\big(\mathsf{Vect}_H^{\omega|_H}\big)$ to$\mathcal{Z}\big(\mathsf{Vect}_G^\omega\big)$ for any subgroup $H$ of $G$ whichpreserves braiding and ribbon structure. As an application, we classify rigidFrobenius algebras in $\mathcal{Z}\big(\mathsf{Vect}_G^\omega\big)$, recoveringthe classification of \'etale algebras in these categories by Davydov-Simmons[J. Algebra 471 (2017), 149-175, arXiv:1603.04650] and generalizing theirclassification to algebraically closed fields of arbitrary characteristic.Categories of local modules over such algebras are modular tensor categories byresults 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.

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