Let $L/\mathbb{Q}$ be a finite Galois extension. Since Galois automorphisms preserve integrality, they also stabilise the ring of integers. When do Galois automorphisms also stabilise all non-maximal orders as well?
A partial answer is given when $L$ is quadratic. In this case, if $\mathcal{O}_L$ is the ring of integers, then any non-maximal order is of the form $\mathbb{Z} + f \mathcal{O}_L$ for some integer $f$ (the conductor). It is obvious now that such orders are preserved by the Galois automorphism.
Yet this need not happen for larger fields. In an answer to this question, a counter-example was exhibited: let $L$ be the splitting field of $X^3 - 3X -1$, which is a cyclic Galois extension of degree $3$. Then one can find an integer $\theta \in \mathcal{O}_L$ such that a Galois conjugate of $\theta$ is $\theta^2/3 - 6$ and prove that this conjugate does not lie in the order $\mathbb{Z}[\theta]$.
Are there other classes of number fields where all orders are preserved?
