Two-transitive groups on a hyperbolic unital

A classification given previously of all projective translation planes of order q 2 that admit a collineation group G admitting a two-transitive orbit of q + 1 points is applied to show that the only projective translation planes of order q 2 admitting a hyperbolic unital acting two-transitively on a secant are the Desarguesian planes and the unital is a Buekenhout hyperbolic unital.