$G$-complete reducibility and saturation

Let $H \subseteq G$ be connected reductive linear algebraic groups defined over an algebraically closed field of characteristic $p> 0$. In our first main theorem we show that if a closed subgroup $K$ of $H$ is $H$-completely reducible, then it is also $G$-completely reducible in the sense of Serre, under some restrictions on $p$, generalising the known case for $G = GL(V)$. Our proof uses R.W. Richardson's notion of reductive pairs to reduce to the $GL(V)$ case. We study Serre's notion of saturation and prove that saturation behaves well with respect to products and regular subgroups. Our second main theorem shows that if $K$ is $H$-completely reducible, then the saturation of $K$ in $G$ is completely reducible in the saturation of $H$ in $G$ (which is again a connected reductive subgroup of $G$), under suitable restrictions on $p$, again generalising the known instance for $G = GL(V)$. We also study saturation of finite subgroups of Lie type in $G$. We show that saturation is compatible with standard Frobenius endomorphisms, and we use this to generalise a result due to Nori from 1987 in case $G = GL(V)$.