  We give a classification of all non-symplectic automorphisms of prime order p
acting on irreducible holomorphic symplectic fourfolds deformation equivalent
to the Hilbert scheme of two points on a K3 surface, for p=2,3 and 7\leq p \leq
19. Our classification relates the isometry classes of two natural lattices
associated to the action of the automorphism on the second cohomology group
with integer coefficients with some invariants of the fixed locus and we
provide explicit examples. As an application, we find new examples of
non-natural non-symplectic automorphisms.
