Let $K$ be a perfect field and suppose $C$ is a geometrically irreducible smooth $K$-curve. Assume that $K(C)/K(t)$ is Galois with $G=Gal(K(C)/K(t))$. Since $C$ is geometrically irreducible, $K$ is algebraically closed in $K(C)$. Let $L/K$ be a Galois extension of $K$ with $H=Gal(L/K)$. We will write $C_L$ to mean $C$ considered as an $L$-curve.
Assuming $L(t)$ is linearly disjoint from $K(C)$ over $K(t)$, we have that the Galois group $Gal(L(C_L)/K(t))$ is isomorphic to $H\times G$.
Suppose $N$ is a subgroup of $Gal(L(C_L)/K(t))$ and let $F=L(C_L)^N$ be the subfield of $L(C_L)$ fixed by $N$. Then $F$ is the function field of some absolutely irreducible $L$-curve $D$, and there is a surjective morphism $\chi:C_L\rightarrow D$. As I understand it, the map $\chi$ identifies points of the curve $C_L$ iff they are in the same $N$-orbit.
My question is, what can we conclude about the map $\chi$ from information about $N$?
For instance if we have that $F\cap L=K$, and so $F$ is the function field of a geometrically irreducible, smooth projective $K$-curve $D$, does it follow that $\chi:C_L\rightarrow D$ is defined over $K$? If $N=H\times\{id_G\}$, this must be the case, since we would have $F=K(C)$.
As a follow up, how does this relate to monodromy and things like that? It seems to me that $Gal(L(C_L)/ K(t))$ is something like an arithmetic monodromy group, though there is the extra factor of $Gal(L/K)$.
Apologies in advance if this isn't a really well formed question; I'm a beginner in this sort of math.