  In this article we study the centralizer of a minimal aperiodic action of a
countable group on the Cantor set (an aperiodic minimal Cantor system). We show
that any countable residually finite group is the subgroup of the centralizer
of some minimal $\mathbb Z$ action on the Cantor set, and that any countable
group is the subgroup of the normalizer of a minimal aperiodic action of an
abelian countable free group on the Cantor set. On the other hand we show that
for any countable group $G$, the centralizer of any minimal aperiodic
$G$-action on the Cantor set is a subgroup of the centralizer of a minimal
$\mathbb Z$-action.
