$\newcommand{\Q}{\mathbb Q}$Suppose I have a finite Galois field extension $K/\Q$ with Galois group $S_n$ (originally I thought of any normal subgroup of $S_n$ for minimal $n$ but this might be too ambitious) and suppose that $m\geq n$ and I have a transitive subgroup $G$ of $S_m$ (acting on $m$ elements) and we have the following containment $S_n\unlhd G$. Can I now (constructively) find a Galois field extension $L/K$ such that $\mathrm{Gal}(L/Q) = G$? By "constructively", I mean I want to adjoin $\Q$ with finite elements that I can also determine by some construction (roots of some polynomials). In fact, I prefer to obtain an irreducible polynomial over $\Q$ with degree $m$ such that $L$ is the splitting field of this polynomial.
If this is not possible for a general $G$, for what $G$ do we know this to be possible (maybe taking a group that is the semidirect product of say $C_2^k$ with $S_n$ or something similar?)
Edit : I thought I add the following condition because the above could still be hard to solve. In my case anyway, I am only dealing with $G$ solvable and I actually know that there is a number field Galois over $\Q$ and with Galois group $G$.