  Many challenging questions about the Grothendieck-Teichmueller group, $GT$,
are motivated by the fact that this group receives the injective homomorphism
(called the Ihara embedding) from the absolute Galois group, $G_Q$, of rational
numbers. Although the question about the surjectivity of the Ihara embedding is
a very challenging problem, in this paper, we construct a family of finite
non-abelian quotients of $GT$ that receive surjective homomorphisms from $G_Q$.
We also assemble these finite quotients into an infinite (non-abelian)
profinite quotient of $GT$. We prove that the natural homomorphism from $G_Q$
to the resulting profinite group is also surjective. We give an explicit
description of this profinite group.
  To achieve these goals, we used the groupoid $GTSh$ of $GT$-shadows for the
gentle version of the Grothendieck-Teichmueller group. This groupoid was
introduced in the recent paper by the second author and J. Guynee and the set
$Ob(GTSh)$ of objects of $GTSh$ is a poset of certain finite index normal
subgroups of the Artin braid group on 3 strands. We introduce a sub-poset $Dih$
of $Ob(GTSh)$ related to the family of dihedral groups and call it the dihedral
poset. We show that each element $K$ of $Dih$ is the only object of its
connected component in $GTSh$. Using the surjectivity of the cyclotomic
character, we prove that, if the order of the dihedral group corresponding to
$K$ is a power of 2, then the natural homomorphism from $G_Q$ to the finite
group $GTSh(K, K)$ is surjective. We introduce the Lochak-Schneps conditions on
morphisms of $GTSh$ and prove that each morphism of $GTSh$ with the target $K$
in $Dih$ satisfies the Lochak-Schneps conditions. Finally, we conjecture that
the natural homomorphism from $G_Q$ to the finite group $GTSh(K, K)$ is
surjective for every object $K$ of the dihedral poset.
