Classification of equality cases in Theorem B

Determine whether, for every prime p>2719 and every finite solvable group G with p^2 dividing |G|, equality k(G)=(49p+451)/60 holds if and only if p\equiv161\pmod{240} and G is isomorphic to the group constructed in the proof of Theorem C.

Background

Theorem B proves the lower bound k(G)\geq(49p+451)/60 for primes p>2719, while Theorem C constructs groups attaining this value for primes p congruent to 161 modulo 240. Thus the paper establishes both a universal asymptotic lower bound and an infinite family of examples attaining it.

Problem E asks for a complete characterization of all equality cases beyond the constructed family. The paper later notes in Remark 2.14 that the proposed characterization is valid for groups of the form G=F2\rtimes H with H a p'-group, but the formulation in Problem E concerns the broader class of finite solvable groups and is presented as unresolved there.

References

Is it true that for $p>2719$, equality holds in Theorem B if and only if $p\equiv 161\pmod{240}$ and $G$ is isomorphic to the group constructed in the proof of Theorem C?

— On the number of conjugacy classes of finite solvable groups of order divisible by $p^2$  (2610.12962 - Keller, 9 Oct 2026) in Problem E, Introduction