When is $H(X)$ compact?

Hofmann and Morris recently showed that if $X$ is a compact space and $H(X)$ the autohomeomorphism group of $X$, with the compact-open topology, is also compact, then $H(X)$ is profinite. They asked whether the converse is true: if $G$ is a profinite group then is there is a compact (preferably, connected) space $X$ such that $H(X) = G$?

We survey what is known about this problem, and give some partial positive answers.