Is every T-group a complete T-group? A T-group is a group $G$ such that if $H$ is a normal subgroup of $G$ and $K$ is a normal subgroup of $H$, then $K$ is a normal subgroup of $G$. Inspired by this, I define a complete T-group to be a group $G$ such that, for all subgroups $H$, $K$, and $L$ of $G$, if $K$ is a normal subgroup of $H$, and $L$ is a normal subgroup of $K$, then $L$ is a normal subgroup of $H$. Certainly, every complete T-group is a T-group. What about the converse? That is, is every T-group a complete T-group as I have defined it? Or is there a counterexample?

Consider $A_4$ in $A_5$.  The former is not a $T$-group, which you can see by looking at the subgroup generated by a double transposition.
The latter is a $T$-group, as it is simple.
That's a counterexample.
Note:  not every subgroup of a $T$-group is a $T$-group (a consequence of what you're asking) .  Every normal subgroup is though.
