Wang defined in Quantum symmetry groups of finite spaces a notion of quantum automorphism group. The application to a finite space of $n$ elements is called the quantum permutation group of $n$ symbols, and its quantum subgroups are called quantum permutation groups of degree $n$. See Quantum permutation groups: a survey by Banica–Bichon–Collins for a survey on this notion. We are interested here in the finite quantum permutation groups, see the classification for $n=4$ in Quantum permutation groups acting on 4 points by Banica–Bichon. Note that finite means finite dimensional as a Hopf $\mathrm C^*$-algebra.
Let $K$ be a finite quantum group. Then $\operatorname{Rep}(K)$ is a unitary integral fusion category. Let us call $K$ weakly group-theoretical if $\operatorname{Rep}(K)$ is so, i.e., if it has (up to equivalence) the same Drinfeld center as a sequence of group extensions. It is an open problem whether there exists an integral (complex) fusion category which is not weakly group-theoretical (see Question 2 of Weakly group-theoretical and solvable fusion categories by Etingof–Nikshych–Ostrik), so in particular, whether there exists a finite quantum group which is not so.
Question: Are the finite quantum permutation groups, weakly group-theoretical?
Note that there exists finite quantum groups which are not quantum permutation groups (the smallest example has dimension $24$; see Finite quantum groups and quantum permutation groups by Banica–Bichon–Natale). Moreover, by the paper of ENO cited above, every finite quantum group of dimension less than $84$ is weakly group-theoretical.
This post was inspired by a question asked by J.P. McCarthy in this talk.
