Hyperovals of when q is even

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 , q 2 ) . In this paper the action of G on points and lines of H ( 3 , q 2 ) is investigated. A construction is given of an infinite family of hyperovals of size 2 ( q 3 − q ) of 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.