  In this paper we study the Bruhat decomposition of not necessarily connected
reductive quasi-split groups $G$ with respect to not necessarily connected
parabolic subgroups. If $G$ is defined over a finite field, we construct a
smooth morphism from the stack classifying $F$-zips with $G$-structure to the
stack classifying the generalized Bruhat cells and study the relation between
the resulting stratifications. We apply these general results to the twisted
orthogonal $F$-zip given by the second De Rham cohomology of a relative
surface, focussing on K3-surfaces.
