Intermediate fields of a transcendental field extension

Question

I have a compact (not finite) group $G$ acting linearly on $\mathbb{C}^n$ and consider $\mathbb{C}(X_1\ldots,X_n)$ as a (non algebraic) field extension of $\mathbb{C}(X_1\ldots,X_n)^G$. Let $\mathbb{F}$ be a maximally transcendent intermediate field, i.e., $\mathbb{C}(X_1\ldots,X_n)$ is an algebraic extension of $\mathbb{F}$. What can one say about the Galois group of this algebraic field extension?