  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.
