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

Representing a profinite group as the homeomorphism group of a continuum