Let $E$ and $F$ be two regular Galois extensions of $\mathbb{Q}(t)$ with group $G$, and let $E_{t_0}$ resp. $F_{t_0}$ be the residue fields corresponding to specializing $t\mapsto t_0\in \mathbb{Q}$. If $\{E_{t_0}\mid t_0\in \mathbb{Q}\}=\{F_{t_0}\mid t_0\in \mathbb{Q}\}$, can we conclude that $E$ and $F$ are the same up to some automorphism of $\mathbb{Q}(t)$ (extended to $E$)? (One certainly has to allow automorphisms here, as e.g.\ $\mathbb{Q}(\sqrt{t})$ and $\mathbb{Q}(\sqrt{t-1})$ of course yield the same set of specializations.)
In other words, does the set of specializations already fix the regular extension $E$ up to $PGL_2$-equivalence? If so, how far can the condition $\{E_{t_0}\mid t_0\in \mathbb{Q}\}=\{F_{t_0}\mid t_0\in \mathbb{Q}\}$ be relaxed? (I'm thinking something like "equality up to finitely many exceptions", or "up to a thin set")