My question is: Let $H,K\leq G$ be two characteristic subgroups and assume $H\leq K$. Do we have $K/H$ is characteristic in $G/H$?
We know that any characteristic subgroup of $G/H$ must be of the form $K/H$ for some characteristic subgroup $K$ of $G$ since any automorphism of $G$ induces an automorphism of $G/H$.
But is the converse true? Or any counterexample?