An infinite family of hyperovals of $Q^{+}(5,q)$, $q$ even

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.