If $G=\mathsf{Aut}_k(F)$ acts on field $F$ algebraic over $k$ then do we have: orbit $G\alpha=\text{ roots of minimal polynomial of }\alpha$?

Question

Let $k\subseteq F$ denote an algebraic field extension and let $\alpha\in F$ having $f\in k[x]$ as its minimal polynomial.

My question:

If $\beta\in F$ is a root of $f$ then does there exists some $\sigma\in\mathsf{Aut}_k(F)$ with $\sigma(\alpha)=\beta$?

I know that the answer is "yes" if the extension is normal but am puzzling whether this condition is necessary or can be weakened.

Thank you in advance for taking notice of this question, and sorry if it is a duplicate.