Automorphisms of hyperkähler manifolds have been studied for many different reasons: construct symplectic quotients or study fixed loci in order to find examples of irreducible symplectic varieties, define maps among different deformation families of HK manifolds, find new examples of Enriques manifolds, that are higher dimensional analogue of Enriques surfaces, and for which Pacienza and Sarti recently proved the Morrison-Kawamata cone conjecture. In the first part of the talk I will present a recent result in collaboration with L. Giovenzana, Onorati and Veniani about symplectic rigidity of hyperkähler manifolds of OG10 type. Then I will show that no Enriques manifolds arise as a quotient of a Laza—Saccà—Voisin manifold by a nonsymplectic automorphism induced from the cubic fourfold. Moreover I will show how to get that no Enriques manifolds can be constructed as free quotients of a manifold of OG10 type constructed as a symplectic resolution of a moduli spaces of sheaves on K3 surfaces by an induced automorphism. This is based on a joint work in progress with Billi, F. and L. Giovenzana.