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First some definitions: Let $cd(G)$ be the set of degrees of irreducible complex characters of the group $G$. Let $X$ be the collection of finite groups with the properties: $\{1\}\in X$. If $G$ is a finite group such that for each $n\in cd(G)$, $G$ has a subgroup $H$ of index $n$ with $H\in X$, then $G\in X$ (so the groups of $X$ are those with subgroups of all indices from character degrees, where these subgroups themselves have the same property).

Let $G$ be a finite group whose complex irreducible character degrees are $1 = f_1 < f_2 < \dots < f_t$. For a character $\chi$ of $G$, let $s(\chi)$ be the $k$ such that if $\psi$ is an irreducible constituent of $\chi$ of smallest possible degree, then $\psi(1) = f_k$. Let $v(\chi)$ be the $k$ such that if $\psi$ is an irreducible constituent of $\chi$ of largest possible degree, then $\psi(1) = k$.

What I am trying to prove (and hoping holds) is that if $G\in X$ and $\chi$ is an irreducible character of $G$ of degree $n = f_k$, then $G$ has a subgroup $H$ of index $n$ such that $H\in X$ and additionally, $$s(\chi_H) + min\{v(\psi^G) | \psi\in Irr(H)\}\leq k$$

The motivation for this is that if I can prove this, then I can prove the Taketa inequality for groups in $X$. Does anyone know if the result holds, or can someone find an example where it does not. It would still be interesting if the result turns out to hold under some more restrictions on $X$, like requiring the existence of subgroups of a larger set of orders, or requiring more structure on those subgroups.

Edit: Disregard this question. I will ask a related but different question which turned out be what really mattered.

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