I know from here that any normal subgroup $K$ of a normal subgroup $H$ in a group $G$ is not necessarily normal in $G$. But I was wondering if this is true in the following case:
Let $G$ be a group. Let $H$ be a normal subgroup of the commutator subgroup $(G,G)$. Is $H$ a normal subgroup of $G$ ?
Thanks in advance for your enlightenment?
K. Y.
Consider $G=A_4$, the alternating group of order $12$. Its derived subgroup is $V_4$, the unique Sylow $2$-subgroup of $A_4$. As $V_4$ is Abelian, any of its $2$-element subgroups $H$ is normal in $V_4$, say $H=\left<(1\,2)(3\,4)\right>$. But $H$ is not normal in $A_4$.