  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.
