Let σ be a semisimple
automorphism of a connected reductive group G, and let Gσ be the fixed
points of σ. We consider the Gσ-orbits on the space of nilpotent elements in
an eigenspace of dσ. We give a desingularization of the orbit closures and
relate the Gσ-orbits to the G-orbits. Along the way, we describe the fixed
points of σ on a flag variety G∕P where P is a σ-stable parabolic subgroup of
G.