Classification of finite-dimensional Nichols algebras of rank two in twisted Yetter--Drinfeld categories

Let $G$ be a finite non-abelian group, let $\mathrm{\Phi }\in Z^3(G,{\mathbb{C}}^{\times })$ be normalized, and let $V,W$ be finite-dimensional simple objects of ${}_G^G{\mathcal{Y}\mathcal{D}}^{\mathrm{\Phi }}$. We classify, up to interchange, the braided-indecomposable pairs $(V,W)$ whose supports generate $G$ and for which $\mathcal{B}(V\oplus W)$ is finite-dimensional. The classification consists of eight cases with five possible support quandles. In every case the Cartan graph is standard of type $A_2$, $B_2$, or $G_2$, and the dimension is determined explicitly. A new phenomenon occurs for ${\mathrm{\Gamma }}_2$: the twisted setting admits a family of type $G_2$ absent from the ordinary ${\mathrm{\Gamma }}_2$ classification and we construct an explicit example over a non-abelian group of order $16$.