Example of a nontransitive action of $\operatorname{Aut}(K/\mathbb Q)$ on the roots in $K$ of an irreducible polynomial.

Question

I want to find an irreducible polynomial $f(x)$ over $\mathbb Q$ and a finite nonnormal extension $K/\mathbb Q$ which contains at least two roots of $f(x)$ such that $\operatorname{Aut}(K/\mathbb Q)$ acts nontransitively on the roots of $f(x)$ in $K$. I could not find simple examples of this.

Update: Julian Rosen gave an answer to the original question above but now I want to impose the condition that $K$ is contained in the splitting field of $f(x)$ because I was looking for intermediate field extensions of the splitting field.

Answer 1

Take $f(x)=x^2-2$, $K=\mathbb{Q}(\sqrt[4]{2})$. Then $K$ contains two roots of $f$, but there is no automorphism exchanging them because only one of the roots is a square in $K$.