Let G be an irreducible collineation group of a finite projective plane π of even order n ≡ 0 mod 4. Our goal is to determine the structure of G under the hypothesis that G fixes a hyperoval Ω of π. We assume |G| ≡ 0 mod 4. If G has no involutory elation, then G = O(G) 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 ∼ = Alt(6) and n = 4, or S/Z(S) ∼ = (C 3 × C 3 ) C 2 , Z(S) is a (possibly trivial) 3-group and n ≡ 1 mod 3. In the latter case there exists a G-invariant subplane π in π such that the collineation group G induced by G on π is irreducible and fixes a hyperoval Ω. Furthermore, the subgroup S generated by all involutory elations in G is a generalized Hessian group of order 18, that is S ∼ = (C 3 × C 3 ) C 2 and the configuration of the centers of the involutory elations in G consists of the nine inflexions of an equianharmonic cubic of a subplane π 0 of order 4. In particular, π 0 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 π with respect to G.