  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 $G$ which
preserves braiding and ribbon structure. As an application, we classify rigid
Frobenius algebras in $\mathcal{Z}\big(\mathsf{Vect}_G^\omega\big)$, recovering
the classification of \'etale 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.
