Is a cover Galois if and only if it is geometrically Galois

If $\pi$ is Galois, then so is $\pi_{\overline{K}}$. This is more or less obvious.

But the converse is false. Consider the cover $X:=\mathbb P^1_{K} \to \mathbb P^1_{K}$, $x\mapsto x^3$. On the level of function fields, it corresponds to the extension $K(t)\subset K(s)$, $s^3=t$. This extension is not Galois if $K=\mathbb Q$, but becomes cyclic over $\overline{\mathbb Q}$.

QiL has given what has to be the simplest example of a geometrically Galois but not Galois extension of function fields. I want to give different -- so, necessarily, less simple -- example, but one that shows how important this distinction is in arithmetic geometry.

Let $K$ be a field -- say of characteristic zero -- let $E_{/K}$ be an elliptic curve, and let $\varphi: E \rightarrow E'$ be an isogeny of degree $n$. Then $C = \operatorname{ker} \varphi$ is a cyclic subgroup of order $n$, and we may view $E' = E/C$. From this one can see that $\varphi$ is an unramified covering map and that the deck transformations are precisely $\tau_P: x \mapsto x + P$ with $P \in C(\overline{K})$.

This means the cover is geometrically Galois: $\# C(\overline{K}) = \operatorname{deg} \varphi$. Moreover, it is Galois iff the Galois action on $C(\overline{K})$ is trivial: i.e., rather than being stabilized by the Galois group of $\overline{K}/K$ (which is necessary for the isogeny $\varphi$ to be defined), in order for $K(E)/K(E')$ to be Galois we need the action of $\operatorname{Gal}(\overline{K}/K)$ to be trivial. In the case of cyclic $C$, this is the distinction between a rationally defined subgroup and a rational torsion point.

The second condition is definitely more stringent than the first. For instance, suppose $K = \mathbb{Q}$ and $C$ is a cyclic subgroup of prime order $p$. Then by work of Mazur, the largest prime $p$ such that there exists an elliptic curve $E$ and a rationally defined subgroup $C$ is $p = 163$, whereas the largest prime such that there exists a rational torsion point of order $p$ is $p = 7$.

Similarly, taking $E$ to be a generic elliptic curve over $K = \mathbb{Q}(t)$, the distinction between geometrically Galois and Galois can be viewed as responsible for the difference between the modular curves $X_1(N)$ and $X_0(N)$.