  We show that Mukai's classification of finite groups which may act
symplectically on a complex K3 surface extends to positive characteristic $p$
under the 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 the
assumption (ii) we classify all possible new groups which may appear. We prove
that the assumption on the order of the group is always satisfied if $p > 11$
and if $p=2,3,5,11$ we give examples of K3 surfaces with finite symplectic
automorphism groups of order divisible by $p$ which are not contained in
Mukai's list.
