I am trying to solve problem (12.4) in Isaacs' Character theory of finite groups.
The problem is to show that if $G$ is non-abelian solvable and has no non-abelian factor group of prime power order, then there is a subgroup $L$ such that $b(L)\le b(G)/r$ and $|G:L|\le b(G)r$ for some integer $r$ such that $2\le r\le b(G)$.
Here $b(H)$ denotes the maximal degree of an irreducible character of the group $H$.
The condition implies that all factor groups $G/O^p(G)$ are abelian, so $G'\subseteq O^p(G)$ for all primes $p$. Also some $O^p(G)<G$ since $G$ is solvable.
That is about as far as I have come, but I don't know if that is relevant.
I have tried to use Cor. (12.8) and Lemma (12.22), but without any success.
Does anyone have any ideas?