$p$-torsion for unramified Artin--Schreier covers of curves

Let $Y\to X$ be an unramified Galois cover of curves over a perfect field $k$ of characteristic $p>0$ with $\mathrm{G}\mathrm{a}\mathrm{l}(Y\mathrm{/}X)\cong \mathbb{Z}\mathrm{/}p\mathbb{Z}$, and let $J_X$ and $J_Y$ be the Jacobians of $X$ and $Y$ respectively. We consider the $p$-torsion subgroup schemes $J_X[p]$ and $J_Y[p]$, analyze the Galois-module structure of $J_Y[p]$, and find restrictions this structure imposes on $J_Y[p]$ (for example, as manifested in its Ekedahl–Oort type) taking $J_X[p]$ as given.