  We construct an infinite family of hyperovals on the Klein quadric
$Q^+(5,q)$, $q$ even. The construction makes use of ovoids of the symplectic
generalized quadrangle $W(q)$ that is associated with an elliptic quadric which
arises as solid intersection with $Q^+(5,q)$. We also solve the isomorphism
problem: we determine necessary and sufficient conditions for two hyperovals
arising from the construction to be isomorphic.
