TITLE: On group quotient by a characteristic subgroup
QUESTION [6 upvotes]: 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?

REPLY [4 votes]: Take $G$ to be the dihedral group $D_4$. It has a presentation $\langle r,s\mid r^4=1=s^2,sr=r^3s\rangle$. Let $K=\langle r\rangle$ the subgroup of rotations (which is thus cyclic of order $4$), and $H=\langle r^2\rangle$. Any automorphism of $G$ must preserve $K$, since it must map $r$ to an element of order $4$, and has a consequence it also fixes $r^2$, thus $K$ and $H$ are characteristic. But $G/H$ is the Klein group, which has no non-trivial characteristic subgroup, and $K/H$ is the cyclic group of order $2$.