On two-transitive parabolic ovals

We examine the state of knowledge on the following problem. Let π be a finite projective plane of odd order n with an oval Ω and let G be a collineation group of π fixing Ω. Assume G fixes a point P on Ω and acts 2-transitively on Ω-{P}. The usual basic question is: what can be said about π, Ω and G?