For this question all groups are finite, and representations are over $\mathbb{C}$. The setup is that we have $N$ a normal subgroup of $G$, with abelian quotient $A$, and an irrep $V$ of $N$ that is invariant under $G$ conjugation. Then we can construct a bilinear form on $A$ valued in $\mathbb{C}^*$ as follows:
For $x,y$ in $A$, lift $V$ to an irrep $V_x$ of $\langle A,x\rangle$. Then conjugating this by $y$ yields a linear twist of this irrep, $V_x\otimes \lambda$, and we evaluate $\lambda(x)$ to get our pairing:$$\langle \langle x,y\rangle\rangle_V\in \mathbb{C}^*$$
This form is alternating, bilinear, and controls the behaviour of characters of $G$ over $V$.
Note in the case when $V=\mu$ is linear, this form is: $$\langle \langle x,y\rangle\rangle_\mu=\mu([y,x])$$
My questions are:
Is there a group cohomological interpretation of this bilinear form?
How does this relate to the other natural piece of cohomological data we obtain from this setup, the class $\alpha_V\in H^2(A,\mathbb{C}^*)$ we obtain via the associated projective representation?
