Normality is not transitive for any pair of nontrivial quotient groups

From Groupprops

Statement

Suppose $A$ and $B$ are (possibly equal) nontrivial groups. Then, there exist groups $H \le K \le G$ such that all the following conditions are satisfied:

$H$ is a normal subgroup of $K$ and the quotient group $K / H$ is isomorphic to $A$ .
$K$ is a normal subgroup of $G$ and the quotient group is isomorphic to $B$ .
$H$ is not a normal subgroup of $G$ .

Related facts

Proof

The construction is as follows. Let $G$ be the wreath product of $A$ and $B$ . Let $K$ be the subgroup $A B$ , i.e., the normal subgroup that forms the base of the semidirect product, and let $H$ be the subgroup of $K$ where a particular coordinate is the identity element. (If we are thinking of $A B$ as functions from $B$ to $A$ , then $H$ can be taken as the subgroup comprising those functions that send the identity element of $B$ to the identity element of $A$ -- here, the particular coordinate becomes the coordinate corresponding to the identity element of $B$ ).

Thus, $K$ is isomorphic to $H \times A$ . Then:

$H$ is normal in $K$ and $K / H$ is isomorphic to $A$ : Fill this in later
$K$ is normal in $G$ and $G / K$ is isomorphic to $B$ : Fill this in later
$H$ is not normal in $G$ : The action of $B$ on the coordinates in $K = A B$ is transitive on the coordinates, so $H$ is not preserved under this action.