This question is about properties of Isom-schemes that are well-known over algebraically closed fields.

Let $K$ be a field of characteristic zero, let $C$ be a smooth projective geometrically connected curve over $K$ and let $P$ a $K$-rational point of $C$. Let $X$ be a smooth projective geometrically connected surface over $K$, and $X\to C$ a curve of genus at least two over $K$. Define $F:= X_P$. Then $F$ is a curve of genus at least two over $K$.

Consider the Isom-scheme:

$\underline{\mathrm{Isom}}_{C}(X, C\times_K F) \to C $.

This map is finite unramified. It is etale if $X\to C$ is "isotrivial", i.e., all geometric fibres of $X\to C$ are isomorphic. This is proven in Deligne-Mumford. (Are there any other references?)

Assume $X\to C$ is isotrivial, **and non-trivial.**

Is the Isom-scheme $\underline{\mathrm{Isom}}_{C}(X, C\times_K F) $ geometrically connected over the ground field $K$? What if $K$ is a number field?