Is there a profinite group $G$, a continuous automorphism $\alpha$ of $G$ and a topologically finitely generated closed subgroup $H \leq G$ such that $\alpha(H) \lneq H$ ?

Note that if an example exists, then $G$ is not topologically finite generated, and $\alpha$ is not given by conjugation by an element of $G$.

If we allow $H$ not to be topologically finitely generated, then the example given by Yves de-Cornullier in Wild automorphisms of a free group gives rise to an example here.