  Given an extension $0\to V\to G\to Q\to1$ of locally compact groups, with $V$
abelian, and a compatible essentially bijective $1$-cocycle $\eta\colon
Q\to\hat V$, we define a dual unitary $2$-cocycle on $G$ and show that the
associated deformation of $\hat G$ is a cocycle bicrossed product defined by a
matched pair of subgroups of $Q\ltimes\hat V$. We also discuss an
interpretation of our construction from the point of view of Kac cohomology for
matched pairs.
  Our setup generalizes that of Etingof and Gelaki for finite groups and its
extension due to Ben David and Ginosar, as well as our earlier work on locally
compact groups satisfying the dual orbit condition. In particular, we get a
locally compact quantum group from every involutive nondegenerate
set-theoretical solution of the Yang--Baxter equation, or more generally, from
every brace structure.
  On the technical side, the key new points are constructions of an irreducible
projective representation of $G$ on $L^2(Q)$ and a unitary quantization map
$L^2(G)\to{\rm HS}(L^2(Q))$ of Kohn--Nirenberg type.
