Non-separability and complete reducibility: $E_n$ examples with an application to a question of K\"ulshammer

Let $G$ be a simple algebraic group of type $E_n(n=6,7,8)$ defined over an algebraically closed field $k$ of characteristic $2$. We present examples of triples of closed reductive groups $H<M<G$ such that $H$ is $G$-completely reducible, but not $M$-completely reducible. As an application, we consider a question of K"ulshammer on representations of finite groups in reductive groups. We also consider a rationality problem for $G$-complete reducibility and a problem concerning conjugacy classes.