Nakajima-extremal Artin--Schreier covers of ordinary elliptic curves in characteristic $2$

Let $k$ be an algebraically closed field of characteristic $2$ and let $q=2^h$, $h\ge 3$. We construct ordinary bielliptic curves $X$ of genus $q+1$ for which $$\text{Aut}(X)\cong \text{Dih}(C_q)\times C_2,\mathrm{\mid }\text{Aut}(X)\mathrm{\mid }=4q=4(g(X)-1)\mathrm{.}$$ These curves realize case \textup{(ib)} in the classification of Giulietti–Korchmáros and give an infinite family answering a problem posed by Korchmáros. More generally, we prove that every curve in case \textup{(ib)} arises from the same construction. Case \textup{(ib)} occurs exactly in genera $g=2^h+1$ with $h\ge 2$. For $g\ge 9$ we determine the full automorphism group; in genus $5$ we determine its Sylow $2$-subgroup but do not claim the full automorphism group. For every fixed $q\ge 8$, the isomorphism classes in case \textup{(ib)} of genus $q+1$ are parametrized bijectively by $(k^{\times }{)}^2$. The construction is described in terms of an ordinary elliptic curve and an invariant differential. We determine the short orbits and ramification, the unramified cyclic quotients, and the quotients by the central involutions. For $q=8$ an explicit plane model over ${\text{F}}_2$ is given.