  Racinet studied the scheme associated with the double shuffle and
regularization relations between multiple polylogarithm values at $N^{th}$
roots of unity and constructed a group scheme attached to the situation; he
also showed it to be the specialization for $G=\mu_N$ of a group scheme
$\mathsf{DMR}_0^G$ attached to a finite abelian group $G$. Then, Enriquez and
Furusho proved that $\mathsf{DMR}_0^G$ can be essentially identified with the
stabilizer of a coproduct element arising in Racinet's theory with respect to
the action of a group of automorphisms of a free Lie algebra attached to $G$.
We reformulate Racinet's construction in terms of crossed products. Racinet's
coproduct can then be identified with a coproduct
$\hat{\Delta}^{\mathcal{M}}_G$ defined on a module $\hat{\mathcal{M}}_G$ over
an algebra $\hat{\mathcal{W}}_G$, which is equipped with its own coproduct
$\hat{\Delta}^{\mathcal{W}}_G$, and the group action on $\hat{\mathcal{M}}_G$
extends to a compatible action of $\hat{\mathcal{W}}_G$. We then show that the
stabilizer of $\hat{\Delta}^{\mathcal{M}}_G$, hence $\mathsf{DMR}_0^G$, is
contained in the stabilizer of $\hat{\Delta}^{\mathcal{W}}_G$. This yields an
explicit group scheme containing $\mathsf{DMR}_0^G$, which we also express in
the Racinet formalism.
