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For even $q$, a group $G$ isomorphic to $PSL(2,q)$ stabilizes a Baer conicinside a symplectic subquadrangle ${\cal W}(3,q)$ of ${\cal H}(3,q^2)$. In thispaper the action of $G$ on points and lines of ${\cal H}(3,q^2)$ isinvestigated. A construction is given of an infinite family of hyperovals ofsize $2(q^3-q)$ of ${\cal H}(3,q^2)$, with each hyperoval having the propertythat its automorphism group contains $G$. Finally it is shown that thehyperovals constructed are not isomorphic to known hyperovals.

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