If $G$ is an infinite compact group, how many orbits can $G$ have under the group action of its continuous automorphisms ?



Action on a compact group
If $G$ is a simple compact matrix Lie group of positive dimension, then there are continuously many orbits: elements with distinct eigenvalues are not conjugate and the outer automorphism group is finite.


If on the other hand $G=\mathbb{Z}\_p$ with $p$ a prime, there are countably many orbits.


On yet another hand, if a compact topological group $G$ is assumed infinite, then I don't think the number of the orbits can be finite: if I remember correctly, $G$ must contain a copy of $S^1$ or the $p$-adic integers, so there will always be countably or continuously many orbits.


Note that if we allow for finite groups, then we can have as many orbits as we wish: if $G=\mathbb{Z}/p^n$ with $p$ a prime, then there are $n+1$ orbits. 


