Non-conjugate subgroups of a finite group have non-isomorphic preimages in some finite extension, so Z-coset equivalent subgroups need not be isomorphic.
Sambale,Characterizing inner automorphisms and realizing outer automorphisms, Advances in Group Theory and Applications22(2025), 155–175
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Subgroups with all finite lifts isomorphic are conjugate
Non-conjugate subgroups of a finite group have non-isomorphic preimages in some finite extension, so Z-coset equivalent subgroups need not be isomorphic.