On the Modular Isomorphism Problem for 2-generated groups with cyclic derived subgroup

We continue the analysis of the modular isomorphism problem for Formula: see text-generated Formula: see text-groups with cyclic derived subgroup, Formula: see text, started in D. García-Lucas, Á. del Río and M. Stanojkowski, On group invariants determined by modular group algebras: Even versus odd characteristic, Algebra Represent. Theory 26 (2022) 2683–2707, doi:10.1007/s10468-022-10182-x. We show that if Formula: see text belongs to this class of groups, then the isomorphism type of the quotients Formula: see text and Formula: see text are determined by its modular group algebra. In fact, we obtain a more general but technical result, expressed in terms of the classification O. Broche, D. García-Lucas and Á. del Río, A classification of the finite 2-generator cyclic-by-abelian groups of prime-power order, Int. J. Algebra Comput. 33(4) (2023) 641–686. We also show that for groups in this class of order at most Formula: see text, the modular isomorphism problem has positive answer. Finally, we describe some families of groups of order Formula: see text whose group algebras over the field with Formula: see text elements cannot be distinguished with the techniques available to us.