  We contribute some information towards finding a general algorithm for
constructing, for a given profinite group, $G$, a compact connected space, $X$,
such that the full homeomorphism group, $H(X)$, with the compact-open topology
is isomorphic to $G$ as a topological group. It is proposed that one should
find a compact topological oriented graph $\Gamma$ such that $G\cong
Aut(\Gamma)$. The replacement of the edges of $\Gamma$ by rigid continua should
work as is exemplified in various instances where discrete graphs were used. It
is shown here that the strategy can be implemented for profinite monothetic
groups $G$.
