Irreducible collineation groups fixing a hyperoval

Let G be an irreducible collineation group of a finite projective plane P of even order n congruent to 0 mod 4. Our goal is to determine the structure of G under the hypothesis that G fixes a hyperoval W of P. We assume |G| congruent to 0 mod 4. If G has no involutory elation, then G = O(G) x S_2 with a cyclic Sylow 2-subgroup S_2 and G has a normal subgroup M of odd order such that a G/M has a minimal normal 3-subgroup. If the subgroup S generated by all involutory elations in G is non-trivial and Z(S) denotes its center, then either S is isomrphic to Alt(6) and n = 4, or S/Z(S) is isomorphic to (C_3 x C_3) x C_2, Z(S) is a (possibly trivial) 3-group and n is congruent to 1 mod 3. In the latter case there exists a G-invariant subplane P_0 in P such that the collineation group G_0 induced by G on P_0 is irreducible and fixes a hyperoval W_0. Furthermore, the subgroup S_0 generated by all involutory elations in G_0 is a generalized Hessian group of order 18, that is S_0 is isomorphic to (C_3 x C_3) x C_2 and the configuration of the centers of the involutory elations in G_0 consists of the nine inflexions of an equianharmonic cubic of a subplane P_1 of order 4. In particular, P_1 is generated by the centers and the axes of all involutory elations in G, and hence it is the so-called Hering's minimal subplane of P with respect to G