  We study the deformed tensor products of complex Hadamard matrices,
$L_{ia,jb}=Q_{ib}H_{ij}K_{ab}$. One problem is that of reconstructing the
quantum group $G_L\subset S_{NM}^+$ out of the quantum groups $G_H\subset
S_N^+,G_K\subset S_M^+$ and of the parameter matrix $Q\in M_{N\times M}(\mathbb
T)$, and we obtain here several results, including complete results in the
Fourier matrix case, $H=F_N,K=F_M$, when the parameter matrix $Q$ is generic.
