According to some form of Tannakian reconstruction, given a finite tensor category with a fiber functor to the category of vector spaces, one determines a Hopf algebra by considering tensor endomorphisms of the fiber functor. As far as I know, a similar procedure is used to reconstruct a group from its symmetric tensor category of representations.
I am curious about what happens if one is given a finite tensor category $mathcal{C}$ and a tensor functor $mathcal{C} to Rep(G)$ for $G$ a finite group. It follows that there should exist a Hopf algebra $H$ (by the previous reconstruction business applied to the composition of this tensor functor with the forgetful functor $Rep(G) to mathrm{Vect}$) and homomorphism $mathbb{C}[G] to H$.
Under what conditions will $H$ be a
semidirect product of $G$ with some Hopf algebra?
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