  We consider the symplectic action of a finite group G on a K3 surface. The
Picard group of the K3 surface has a primitive sublattice determined by G. We
show how to compute the rank and discriminant of this sublattice. We then
describe moduli spaces of K3 surfaces with symplectic G-action, extending
results of Nikulin in the abelian case. We use our moduli spaces to develop
techniques for classifying all possible symplectic actions of a group G.
