We prove that we have an isomorphism of type $A_{aut}(\mathbb
C_\sigma[G])\simeq A_{aut}(\mathbb C[G])^\sigma$, for any finite group $G$, and
any 2-cocycle $\sigma$ on $G$. In the particular case $G=\mathbb Z_n^2$, this
leads to a Haar-measure preserving identification between the subalgebra of
$A_o(n)$ generated by the variables $u_{ij}^2$, and the subalgebra of
$A_s(n^2)$ generated by the variables $X_{ij}=\sum_{a,b=1}^np_{ia,jb}$.

We consider several orthogonal quantum groups satisfying the easiness
assumption axiomatized in our previous paper. For each of them we discuss the
computation of the asymptotic law of Tr(u^k) with respect to the Haar measure,
u being the fundamental representation. For the classical groups O_n, S_n we
recover in this way some well-known results of Diaconis and Shahshahani.