Irreducible collineation groups with two orbits forming an oval

Let G be a collineation group of a finite projective plane π of odd order fixing an oval Ω. We investigate the case in which G has even order, has two orbits Ω0 and Ω1 on Ω, and the action of G on Ω0 is primitive. We show that if G is irreducible, then π has a G-invariant desarguesian subplane π0 and Ω0 is a conic of π0.