Symplectic Automorphisms on Kummer Surfaces

Nikulin proved that the isometries induced on the second cohomology group of a K3 surface $X$ by a finite abelian group $G$ of symplectic automorphisms are essentially unique. Moreover he computed the discriminant of the sublattice of $H^2(X, \Z)$ which is fixed by the isometries induced by $G$. However for certain groups these discriminants are not the same of those found for explicit examples. Here we describe Kummer surfaces for which this phenomena happens and we explain the difference.