Professor Cayley seems the first who found a symmetry of infinite order of a K3 surface. First, we discuss the automorphism he found and the automorphism group of Cayley's K3 surface also in a modern view after Festi Garbagnati, van Geemen and van Luijk and myself.  Then I would  like to describe the full (birational) automorphism group of the Hilbert scheme of lenghts two of a Cayley K3 surface - a compact hyperk/"ahler manifold of dimension four - which drastically changes from the automorphism group of Cayley's K3 surface. Applying a similar method, I will also present an explicit example in which MDS structure is not preserved under crepant extremal resolution.