  We show that under standard assumptions on the isotropy groups of an integer
GKM manifold, the equivariant Stiefel-Whitney classes of the action are
determined by the GKM graph. This is achieved via a GKM-style description of
the equivariant cohomology with coefficients in a finite field $\mathbb Z_{p}$
even though in this setting the restriction map to the fixed point set is not
necessarily injective. This closes a gap in our argument why the GKM graph of a
$6$-dimensional integer GKM manifold determines its nonequivariant
diffeomorphism type. We introduce combinatorial Stiefel-Whitney classes of GKM
graphs and use them to derive a nontrivial obstruction to realizability of GKM
graphs in dimension $8$ and higher.
