  Let $k$ be a nonperfect separably closed field. Let $G$ be a (possibly
non-connected) reductive group defined over $k$. We study rationality problems
for Serre's notion of complete reducibility of subgroups of $G$. In our
previous work, we constructed examples of subgroups $H$ of $G$ that are
$G$-completely reducible but not $G$-completely reducible over $k$ (and vice
versa). In this paper, we give a theoretical underpinning of those
constructions. To illustrate our result, we present a new such example in a
non-connected reductive group of type $D_4$ in characteristic $2$. Then using
Geometric Invariant Theory, we generalize the theoretical result above
obtaining a new result on the structure of $G(k)$-(and $G$-) orbits in an
arbitrary affine $G$-variety. We translate our result into the language of
spherical buildings to give a topological viewpoint. A problem on centralizers
of completely reducible subgroups and a problem concerning the number of
conjugacy classes are also considered.
