Bruhat strata and F-zips with additional structures

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.