What is the smallest finite group $ G $ that is neither solvable-by-perfect $$ 1 \to G_{solvable} \to G \to G_{perfect} \to 1 $$ nor perfect-by-solvable $$ 1 \to G_{perfect} \to G \to G_{solvable} \to 1 $$
Obviously if $ G $ is perfect or solvable then it can be decomposed in both ways. The smallest group that is neither perfect nor solvable is $ S_5 $, but $ S_5=A_5.2 $ is perfect-by-solvable.
Context: A lot of my intuition comes from Lie groups. Every Lie algebra has a maximal solvable radical. Using this, any Lie group $ G $ can be written as $$ 1 \to G_{solvable} \to G \to G_{semisimple} \to 1 $$ in some ways semisimple Lie groups are like perfect finite groups (this analogy is explored in Commutator of a connected semi simple Lie group and in Is the fundamental group of a connected semisimple Lie group equal to its Schur multiplier?). For Lie groups every group is solvable-by-semisimple, using the Levi decomposition. I was wondering how much quickly this idea of decomposing the group into a perfect and solvable part fails for finite groups.
Update, thanks to sTertooy (the comment I left below):
"oh I see. Another way to describe this is that the derived series of a finite group always terminates in a perfect group $ G^{(n)} $ for some minimal $ n $ (see https://en.wikipedia.org/wiki/Commutator_subgroup for example where this fact is explicitly stated) indeed this $ G^{(n)} $ is the $ N $ you describe. Then $ G^{(n)}=N $ is normal because commutator subgroups are always characteristic and a characteristic subgroup of a characteristic subgroup is itself characteristic in the full group. Quotienting by $ G^{(n)}=N $ always gives a solvable group of derived length $ n $"
Summary of what I learned:
Every finite group is perfect-by-solvable because the derived series of a finite group always terminates by finiteness. In general, I think an abstract group is perfect-by-solvable if and only if the derived series terminates (in other words there exists an $ n $ such that $ G^{(n)}=G^{(N)} $ for all $ N \geq n $).
Many finite groups are not solvable-by-perfect, for example $ S_5 $ (the only solvable normal subgroup is $ 1 $ which does not have a perfect quotient).
Every Lie group is solvable-by-semsimple by the Levi decomposition.
Many Lie groups are not semisimple-by-solvable, for example any $ SE(n,\mathbb{R}) \cong \mathbb{R}^n \rtimes SO(n,\mathbb{R}), n \geq 3 $ since this Lie group is not solvable but its only semisimple normal subgroup is the trivial group.
