Let $q=p^n$, for $p$ a prime. Let $C$ be an Artinâ€“Schreier curve defined by $y^p-y=f(x)$ where $f(x) \in \mathbb{F}_{q}[x]$.

Let $C_g$ be $y^p-y=f(x)$ where $x \in g(\mathbb{F}_{p^{n}})$ for some $g \in \mathbb{F}_q[x]$.

Is there a relation between number of solutions of $C$ and $C_g$? Is there a Zeta function for $C_g$ and $C$ ?