Is there an infinite $T_2$-space $X$ with $X\cong \text{Aut}(X)$? (Here, $\text{Aut}(X)$ is the set of automorphisms $\varphi:X\to X$ and it carries the topology inherited from the product topology on $X^X$.)

What's the answer if we endow $\text{Aut}(X)$ with the compact-open topology instead of the product topology?