On Shumyatsky's Question on Soluble Subgroups and the Generalized Fitting Height: A Partial Answer for Groups with Chief Factors of Type PSL(2,q) and Sz(q)

Let h*(G) be the generalized Fitting height of a finite group G, and h(K) the Fitting height of a soluble group K. P. Shumyatsky asked (Kourovka Notebook, Problem 19.82) whether every finite group G has a soluble subgroup K with h*(G) = h(K), that is, whether h*(G) never exceeds the largest Fitting height s(G) of a soluble subgroup; Khukhro and Shumyatsky proved λ(G) ≤ s(G) for the nonsoluble length λ(G), which is at most h*(G) (Israel J. Math. 2015). We give a partial answer: yes for every finite group whose nonabelian chief factors are of type PSL(2,q) (q ≥ 4) or Sz(q), provided that no PSL(2,q)-factor with q odd and q ≡ ±1 (mod 8) induces a group containing PGL(2,q); the general question stays open. For these groups, if O_p(G) = 1, some soluble K has h(K/O_p(K)) ≥ h*(G). Extending the method of Khukhro and Shumyatsky, we exclude a minimal counterexample by a witness system, a condition on the automorphisms induced on a minimal normal subgroup S^m, and construct witness systems for PSL(2,q) and Sz(q) from unipotent subgroups, Zsigmondy primes and the Lang–Steinberg theorem. The excluded case is a limit of the method, not a counterexample; the criterion is only sufficient and fails for some simple groups; A_n (n ≥ 7), the other groups of Lie type and the sporadic groups are not treated. No printed answer to the question was found. This is an unrefereed note. Unrefereed preprint released for independent mathematical scrutiny. Publication on Zenodo does not constitute peer review. AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text. Corpus identifier: KOU-19.82 (Kourovka Notebook, Problem 19.82, P. Shumyatsky, 19th issue, 2018).