Non-conjugate subgroups of a finite group have non-isomorphic preimages in some finite extension, so Z-coset equivalent subgroups need not be isomorphic.
Cornulier,Is every finite group the outer automorphism group of a finite group?, MathOverflow,https: //mathoverflow.net/q/372563
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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.