Classification of modular-invariant boundary maps

Let $G$ be a finite group, let $e$ be its identity element, and let $B:G\times G\to\mathds{C}^{\times}$ be a bivariate map.

Boundary-map classification conjecture. All maps $B$ satisfying

$$\sum_{g'\in G}B(g_1,g')B(g_1,g'^{-1}g_2)=|G|B(g_1,g_2),$$ $$B(e,g)=B(g,e)=1,\qquad B(g_1,g_2)=B(g_2,g_1^{-1}),\qquad B(g_1,g_2)=B(g_1,g_1g_2)$$

are classified by a pair $(H\subseteq G,\omega\in H^2(H,\mathds{C}^{\times}))$. This would extend the explicit Abelian classification to non-Abelian subgroups and identify all such boundary terms with subgroup and 2-cohomology data.