When a solvable group is supersolvable?

Question

I am thinking about the following question:

We know if group $G$ has a normal subgroup such that $N$ and $G/N$ are solvable, then $G$ is solvable. But this is not true for the supersolvable groups.

Question: If $G$ is a solvable group with normal subgroup $N$ such that $N$ and $G/N$ are supersolvable, then $G$ is supersolvable. Is it true? or Is there any counterexample?

If $G/N$ is a small group such as the symmetric group of degree $3$, then it can be true.

Answer 1

Let $G=A_4$ and $N=V_4$, the Kleinian $4$-group (it is the only nontrivial, proper normal subgroup of $A_4$). Then $N$ and $G/N=A_4/V_4\cong C_3$ are supersolvable, but $A_4$ is not supersolvable. In fact, $A_4$ is the smallest example of a finite non-supersolvable group.

On the other hand, there are conditions, so that the supersolvability of $N$, $G/N$ implies that $G$ is supersolvable. For example, if $N$ is cyclic.

Reference: Let $ G $ be a group and $ N \lhd G $. Why is $G $ supersoluble if $ N $ is cyclic and $ G/N $ is supersoluble?