Groups for which all normal subgroups are perfect I am trying to understand the following property of groups. 
A group $G$ is perfect if $G=[G,G]$.
So call $G$ extra-perfect if every normal subgroup of $G$ is perfect. 
One obvious class of extra-perfect groups are the nonabelian simple groups. Are there other examples? Is there some kind of classification, or equivalent characterization of extra-perfection for finite groups?
 A: A finite group $G$ is extra perfect if and only if every composition factor of $G$ is a non-abelian simple group.
Although not at all obvious, this actually isn't too hard to prove using one fact:

Each chief factor of a group $G$ is a direct product of isomorphic simple groups.

A composition factor is just a simple normal subgroup of a chief factor, so if there is an abelian composition factor, then there is an abelian chief factor. That is there are some normal subgroups $N,M$ of $G$ with $N<M\le G$ such that $M/N$ is abelian. But then $[M,M]\le N$ so $M$ is not perfect.
Conversely if every composition factor is non-abelian, let $N\trianglelefteq G$. Take some $M\trianglelefteq G$ maximally properly contained in $N$ (so $M<N$ and if $M\le K<N$ with $K\trianglelefteq G$ then $K=M$). $N/M$ is a chief factor of $G$, so is a direct product of non-abelian simple groups. It is easy to check $N/M$ is perfect so the image of $[N,N]$ in $N/M$ is $N/M$. We can apply induction to assume $M$ is perfect, so $M\le [N,N]$. It's kind of clear here that $N$ is therefore perfect, but lets spell it out:
$$|[N,N]|=|[N,N]/M||M|=|N/M||M|=|N|$$
So $[N,N]=N$
