Automorphisms of O'Grady's Manifolds Acting Trivially on Cohomology

We determine the subgroup of automorphisms acting trivially on the second integral cohomology for hyperkaehler manifolds which are deformation equivalent to O'Grady's sporadic examples. In particular, we prove that this subgroup is trivial in the ten dimensional case and isomorphic to $(\mathbb{Z}/2\mathbb{Z})^{\times 8}$ in the six dimensional case. Furthermore we study the fixed loci of the automorphisms in the six dimensional case.