Hyperovals of $H(3,q^2)$ when $q$ is even

For even $q$, a group $G$ isomorphic to $PSL(2,q)$ stabilizes a Baer conic inside a symplectic subquadrangle $\mathcal{W}(3,q)$ of $\mathcal{H}(3,q^2)$. In this paper the action of $G$ on points and lines of $\mathcal{H}(3,q^2)$ is investigated. A construction is given of an infinite family of hyperovals of size $2(q^3-q)$ of $\mathcal{H}(3,q^2)$, with each hyperoval having the property that its automorphism group contains $G$. Finally it is shown that the hyperovals constructed are not isomorphic to known hyperovals.