A note on Galois modules and the algebraic fundamental group of projective curves

Let $X$ be a smooth projective connected curve of genus $g\ge 2$ defined over an algebraically closed field $k$ of characteristic $p>0$. Let $G$ be a finite group, $P$ a Sylow $p$-subgroup of $G$ and $N_G(P)$ its normalizer in $G$. We show that if there exists an \'etale Galois cover $Y\to X$ with group $N_G(P)$, then $G$ is the Galois group wan \'etale Galois cover $\mathcal{Y}\to\mathcal{X}$, where the genus of $\mathcal{X}$ depends on the order of $G$, the number of Sylow $p$-subgroups of $G$ and $g$. Suppose that $G$ is an extension of a group $H$ of order prime to $p$ by a $p$-group $P$ and $X$ is defined over a finite field $\mathbb{F}_q$ large enough to contain the $|H|$-th roots of unity. We show that integral idempotent relations in the group ring $\mathbb{C}[H]$ imply similar relations among the corresponding generalized Hasse-Witt invariants.