  We study algebraic and topological properties of the convolution semigroups
of probability measures on a topological groups and show that a compact
Clifford topological semigroup $S$ embeds into the convolution semigroup $P(G)$
over some topological group $G$ if and only if $S$ embeds into the semigroup
$\exp(G)$ of compact subsets of $G$ if and only if $S$ is an inverse semigroup
and has zero-dimensional maximal semilattice. We also show that such a Clifford
semigroup $S$ embeds into the functor-semigroup $F(G)$ over a suitable compact
topological group $G$ for each weakly normal monadic functor $F$ in the
category of compacta such that $F(G)$ contains a $G$-invariant element (which
is an analogue of the Haar measure on $G$).
