### Abstract

We show that Mukai’s classification of finite groups which may act symplectically on a complex K3 surface extends to positive characteristic $p$ under assumptions that (i) the order of the group is coprime to $p$ and (ii) either the surface or its quotient is not birationally isomorphic to a supersingular K3 surface with Artin invariant 1. In the case without assumption (ii) we classify all possible new groups which may appear. We prove that assumption (i) on the order of the group is always satisfied if $p > 11$. For $p=2,3,5,11$, we give examples of K3 surfaces with finite symplectic automorphism groups of order divisible by $p$ which are not contained in Mukai’s list.