  We show that every bounded automaton group can be embedded in a finitely
generated, simple amenable group. The proof is based on the study of the
topological full groups associated to the Schreier dynamical system of the
mother groups. We also show that if $\mathcal{G}$ is a minimal \'etale groupoid
with unit space the Cantor set, the group $[[\mathcal{G}]]_t$ generated by all
torsion elements in the topological full group has simple commutator subgroup.
