Hyperovals of H(3,q²) when q is even

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For even q, a group G isomorphic to PSL(2,q) stabilizes a Baer conic inside a symplectic subquadrangle W(3,q) of H(3,q²). In this paper the action of G on points and lines of H(3,q²) is investigated. A construction is given of an infinite family of hyperovals of size 2(q³-q) of H(3,q²), 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.