Is the automorphism group of a compact (metric) space locally compact?

I am interested in finding out whether or not the automorphism group of a compact topological space $X$ (with appropriate topology e.g. 'weak' or compact-open) is a locally compact space.

Or, what extra conditions can we put on $X$ to ensure that $Aut(X)$ is locally compact?... What if $X$ is, say, a metric space, i.e. when is the isometry group locally compact?

When is the group of homeomorphisms of a compact space locally compact?

I am interested in finding out when the group of homeomorphisms of a compact topological space $X$ (with appropriate topology e.g. 'weak' or compact-open) is a locally compact space.

What extra conditions might we be able to put on $X$ to ensure that it is so?... What if $X$ is, say, a metric space and we ask when the isometry group is locally compact?