Is the action of G on X necessarily transitive?

Question

Note: The question first proposed here considering $X$ to be the general notion of a topological space and thanks to a respected answer it is resolved for the general case but an answer is still demanding for mentioned condition of $X$ below.

$G$ is a group of homeomorphisms from a compact connected Hausdorff topological space $X$ to itself with uncountable number of elements and we have $|G|= |Aut(X)|$.

Question: Is the action of G on X necessarily transitive?

Answer 1

Here's an easy counterexample to the strongest version mentioned so far: given a cardinal $\kappa$, consider the "disjoint union" $X$ of a discrete space of size $\kappa$ and an indiscrete space of size $2$. Then $Aut(X)$ has cardinality $2^{\vert X\vert}$ but clearly isn't transitive (it doesn't move anything "between the pieces").