This is a follow-up of a sort to a question I asked earlier, but this turned out to be what was really interesting, and the main relation between the questions is that they share some definitions, which I will mention again here anyway, and that the end result is the Taketa inequality.
First some definitions: let $cd(G)$ be the set of degrees of irreducible complex characters of the group $G$. If these degree are $1 = f_1 < f_2 < \dots < f_t$ and $\chi$ is a character of $G$ with $\psi$ an irreducible constituent of $\chi$ of lowest possible degree and $\varphi$ an irreducible constituent of $\chi$ of largest possible degree, such that $\psi(1) = f_n$ and $\varphi(1) = f_m$, then define $s(\chi) = n$ and $v(\chi) = m$. Let $G$ be a finite group, let $\chi$ be an irreducible character of $G$ of degree $f_k$ and let $H$ be a subgroup of $G$. We call $(G,H,\chi)$ a good triple if there is an irreducible character $\psi$ of $H$ such that $s(\chi_H) + v(\psi^G) \leq k$.
Now define the collection of finite group $X$ by the property that a finite group $G$ is in $X$ if and only if either $G$ is an $M$-group, or if for each non-linear irreducible character $\chi$ of $G$, there is a proper subgroup $H$ of $G$ such that $H\in X$ and $(G,H,\chi)$ is a good triple.
I am interested in what "nice" properties of a group will guarantee that it is in this collection. I have shown that nilpotent groups and groups of squarefree order are in $X$, but I would like to find some groups in $X$ which are not $M$-groups, since I can prove that the Taketa inequality holds for all groups in $X$, and this is of course not really interesting unless $X$ contains groups other than $M$-groups
Edit: Changed the definition a bit to include all $M$-groups, since I found out that I can still prove the same things about these groups, and the collection becomes considerably larger this way.
Edit2: I have found that knowing the character degrees of the derived subgroup is often enough to determine the existence of good triples, namely via the following result:
If $\chi\in Irr(G)$ with $\chi(1) = f_k$ (with $f_k$ as above) then define $pos(\chi) = k$ (the position of $\chi$). If $\chi$ is not linear and $(G,G',\chi)$ is not a good triple, then there is a character $\psi\in Irr(G')$ such that $pos(\psi)\geq pos(\chi)$ and $\psi(1)$ divides $\chi(1)$ (actually, such that $\psi(1)t$ divides $\chi(1)$ where $t = |G:I_G(\psi)|$)