  The aim of this paper is to show the rigidity of homologically trivial
actions of prime order on K3 surfaces. To be precise, we show that homotopy K3
surfaces do not admit a periodic diffeomorphism of odd prime order 3 acting
trivially on cohomology. Moreover, we give an obstruction in terms of the
rationality and sign of the spin numbers to the existence of a periodic
diffeomorphism of odd prime order acting trivially on cohomology of homotopy K3
surfaces.
