 Solution

Artin fixed-field theorem says that for a finite group $G\le \mathrm{Aut}L$, $L/L^G$ is Galois with group $G$ and degree $|G|$. Let any finite $G$ act faithfully by left translation on variables $X_g$ in $L=\mathbb{C}(X_g:g\in G)$. Artin's theorem then realizes $G$ as $\mathrm{Gal}(L/L^G)$.