Galois action on prime ideals

Question

Suppose $A$ is a $k$-algebra, with $k$ a field, and let $\overline{A} := A \otimes_k \overline{k}$, where $\overline{k}$ is an algebraic closure of $k$. Let $\frak{p}$ be any prime ideal in $A$, and let $S(\frak{p})$ be the set of prime ideals in $\overline{A}$ over $\frak{p}$. My question is the following: does $\mathrm{Aut}(\overline{k}/k)$ act transitively on $S(\frak{p})$? I know the answer is "yes" if "prime ideal" is replaced by "maximal ideal," but am interested in the more general setting. (I have also seen variations of this question in the context of, e.g., Dedekind domains, but I found nothing yet about my specific setting.)