  In this paper, we show that the action of a characteristically simple,
non-extremely amenable (non-strongly amenable, non-amenable) group on its
universal minimal (minimal proximal, minimal strongly proximal) flow is
effective. We present necessary and sufficient conditions, for the action of a
topological group with trivial center on its universal minimal proximal flow,
to be effective. A theorem of Furstenberg about the isomorphism of the
universal minimal proximal flows of a discrete group and its subgroups of
finite index ([Theorem~II.4.4]) is strengthened. Finally, for a pair of groups
$H < G$ the same method is applied in order to extend the action of $H$ on its
universal minimal proximal flow to an action of its commensurator group
$\mathrm{Comm}_G(H)$.
