Minimal Non-F-Groups with Sylow Residual and the Unitary Groups PSU(3, p²ⁿ): A Negative Answer to Kourovka Notebook Problem 15.98

In Problem 15.98 of the Kourovka Notebook (2002), L. A. Shemetkov asked whether, for a saturated formation F and a finite soluble minimal non-F-group G whose F-residual is a Sylow p-subgroup, this residual is always isomorphic to a factor group of the Sylow p-subgroup of PSU(3, p²ⁿ). For the formation of nilpotent groups this is a theorem of Mazurov and Syskin (p = 2) and of Zhurtov and Syskin on Schmidt groups. We show that the answer is no: there is a saturated formation F and a finite soluble minimal non-F-group G of order 2304 whose F-residual is a Sylow 2-subgroup isomorphic to no factor group of a Sylow 2-subgroup of PSU(3, 2²ⁿ); 2304 is the least order of such a counterexample. The obstruction is elementary: in a nonabelian factor group Q of the Sylow p-subgroup of PSU(3, p²ⁿ) the commutators [y, x], x ∈ Q, fill Q' for every non-central y, while our residual, a special group of order 2^8 with centre of order 4, has a non-central element with only two commutators. The example comes from an elementary construction, a cyclic group acting on a special p-group, which gives a second counterexample of order 3^8·2^4 and shows that the answer stays no for subgroup-closed formations containing all supersoluble groups, for formations contained in N_p N_r, and for the formations of metanilpotent and of nilpotent-by-abelian groups. For the formation of supersoluble groups the answer is yes; we include a proof, not claimed as new, which uses the theorem of Zhurtov and Syskin as stated in its review. The least-order statement does not use that theorem. No printed answer to the question was found in the sources we searched; a search that finds nothing is not a proof of novelty, and the construction may be known to specialists. 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-15.98 (Kourovka Notebook Problem 15.98, L. A. Shemetkov, 2002; UnsolvedMath v1.8.0).