Tambara-Yamagami category is a fusion category which its simple objects are elements of a group and one element out of group. i.e : $$simple\;objects = G \cup \{m\}$$ The fusion rule of this category is : $$m\times m=\sum_{i \in G} g_i, \hspace{0.5cm} g_i\times g_j=g_i*g_j(*:group\;action),\hspace{0.5cm} m\times g_i=g_i\times m =m $$ As far as I know, according to [1],[2] there is a complete classification for braided Tambara-Yamagami category if we put G an abelian (2-)group.
My question is about this category if G is a non-abelian group.
1-Tambara, Daisuke, and Shigeru Yamagami. "Tensor categories with fusion rules of self-duality for finite abelian groups." Journal of Algebra 209.2 (1998): 692-707.
2-Siehler, Jacob A. "Braided near-group categories." arXiv preprint math/0011037 (2000).