I am self reading from Groups as Galois Group by Helmut Volklein There is a result on page 94(section 5.4)

Let $G$ be a finite group. Let $P\subset P^{1}$ finite and $q\in P^{1}\P$. There is a natural $1-1$ correspondence between

- The $\mathbb{C}(x)$-isomorphism classes of Galois extensions $L/\mathbb{C}(x)$ with Galois group isomorphic to $G$ and with branch points contained in $P$.
- The equivalence class of Galois coverings $f:R\rightarrow P^{1}\P$ with deck transformation group isomorphic to $G$.
- The normal subgroups of the fundamental group $\pi_{1}(P^{1}\p,q)$ with quotient isomorphic to $G$.

I am interested in Inverse Galois problem over $\mathbb{Q}(T)$ as then by Hilbert's Irreducibility theorem I can study IGP over $\mathbb{Q}$

Is there similar theorem(result) that states the correspondence between field extensions of $\mathbb{Q}(T)$ and coverings of $P^{1}_{Q}$.

**Edits**

**Why I think there is such result**

See the attached image [ Topics in Galois Theory, by Jean-Pierre Serre]

He also mentions the same idea, He has written the statement but I want to convince myself (mathematically by a proof) that it is actually true.