19.51 (2018)

A finite group is monomial if each irreducible character of it is induced from a linear character of some subgroup. Monomial groups are soluble (Taketa). The group is normally (subnormally) monomial if each irreducible character is induced from a linear character of some normal (respectively, subnormal) subgroup. Metabelian groups are normally monomial, and abelian-by-nilpotent groups are subnormally monomial. In the other direction, it is known that there exist normally monomial groups of arbitrarily large derived length.
$\qquad$ a) For a given prime $p$, do there exist normally monomial finite $p$-groups of arbitrarily large derived length?
$\qquad$ b) Do there exist subnormally monomial groups of arbitrarily large nilpotence length?