Let $k$ be an arbitrary field and $G$ be an arbitrary finite group. It seems to me that one can construct fields $F$ and $E$ such that $k\subset F\subset E$ and $\rm Gal$$(E/F)$ is isomorphic to $G$ as follows:
By Cayley's theorem, $G$ can be embedded into a symmetric group $S_n$ for some integer $n$. Let $E$ be the quotient field of polynomial ring $k[X_1,X_2,...,X_n]$ where $X_1, ...,X_n$ are $n$ independent variables. Then there is a natural isomorphism between $S_n$ and the subgroup $H$ of $\rm Gal$$(E/k)$ such that any $\sigma\in H$ permutes the set $\{X_1, ..., X_n\}$. Let $G_0$ denote the subgroup of $H$ isomorphic to $G$, and denote $E^{G_0}$ by $F$. Since $G_0$ is finite, by Artin's throrem we know $E$ is Galois over $F$ and $G_0=$$\rm Gal$$(E/F)$.
It seems that there are something not correct in this proof but I cannot tell which part is wrong. Can anyone help?