Ekedahl-Oort types of $\mathbb{Z}/2\mathbb{Z}$-covers in characteristic $2$

In this article we study the Ekedahl-Oort types of $\mathbb{Z}/2\mathbb{Z}$-Galois covers $\pi:Y \to X$ in characteristic two. When the base curve $X$ is ordinary, we show that the Ekedahl-Oort type of $Y$ is completely determined by the genus of $X$ and the ramification of $\pi$. For a general base curve $X$, we prove bounds on the Ekedahl-Oort depending on the Ekedahl-Oort type of $X$ and the ramification of $\pi$. Along the way, we develop a theory of \emph{enhanced differentials of the second kind}. This theory allows us to study algebraic de Rham cohomology in any characteristic by working directly with differentials, in contrast to the standard \v{C}ech resolution.