From projective representations to pentagonal cohomology via quantization

Given a locally compact group $G=Q\ltimes V$ such that $V$ is Abelian and such that the action of $Q$ on the Pontryagin dual $\hat V$ has a free orbit of full measure, we construct a family of unitary dual $2$-cocycles $\Omega_\omega$ (aka non-formal Drinfel'd twists) whose equivalence classes $[\Omega_\omega]\in H^2(\hat G,\mathbb T)$ are parametrized by cohomology classes $[\omega]\in H^2(Q,\mathbb T)$. We prove that the associated locally compact quantum groups are isomorphic to cocycle bicrossed product quantum groups associated to a pair of subgroups of the dual semidirect product $Q\ltimes\hat V$, both isomorphic to $Q$, and to a pentagonal cocycle $\Theta_\omega$ explicitly given in terms of the group cocycle $\omega$.