  For even $q$, a group $G$ isomorphic to $PSL(2,q)$ stabilizes a Baer conic
inside a symplectic subquadrangle ${\cal W}(3,q)$ of ${\cal H}(3,q^2)$. In this
paper the action of $G$ on points and lines of ${\cal H}(3,q^2)$ is
investigated. A construction is given of an infinite family of hyperovals of
size $2(q^3-q)$ of ${\cal H}(3,q^2)$, 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.
