In this question, let us call a finite dimensional $\mathbb{C}^*$ Hopf algebra $H$ strongly group theoretical if there exists a finite group $G$ such that one of the following three equivalent statements is true:
(1). The Drinfeld double $D(H)$ is twist equivalent to $D(G)$, i.e., there exists a Drinfeld twist $J\in D(G)\otimes D(G)$ such that the twisted Hopf algebra $D(G)^J$ is isomorphic to $D(H)$;
(2). The Drinfeld centers $Rep[D(H)]=\mathcal{Z}[Rep(H)]$ and $Rep[D(G)]=\mathcal{Z}[Rep(G)]$ are equivalent as braided fusion categories;
(3). The fusion categories $Rep(G)$ and $Rep(H)$ (the category of representations of $G$ and $H$, resp.), are Morita equivalent (see also this question).
Question: what is the smallest $\mathbb{C}^*$-Hopf algebra that is not strongly group theoretical? And how to systematically construct this type of $\mathbb{C}^*$-Hopf algebras?
Remarks and updates:
The smallest non-commutative non-cocommutative (ncnc) $\mathbb{C}^*$-Hopf algebra is the 8-dimensional Kac-Paljutkin Hopf algebra $H_8$, which is certainly not isomorphic to any group algebra. However, $H_8$ is twist equivalent to the group algebra $\mathbb{C}[D_4]$, where $D_4$ is the dihedral group $D_4 = \langle a, b: a^4 = b^2 = e, a b = b a^{-1} \rangle $, and consequently, $D(H_8)$ is twist equivalent to $D(D_4)$, see here. The next ncnc $\mathbb{C}^*$-Hopf algebras come in dimension 12, there are two, both of which are self dual, triangular, and twist equivalent to group algebras, see this paper. These are the only three ncnc $\mathbb{C}^*$-Hopf algebras up to dimension 15.
For $H$ to be not strongly group theoretical, both $H$ and its dual $H^*$ must not be twist equivalent to any group algebra. The smallest such $\mathbb{C}^*$-Hopf algebras come in dimension 16, listed in Table I of Kashina (2000). Is anyone of them not strongly group theoretical?
