For finite abelian p-groups, for 2-groups of maximal class, and for 2-groups of order at most 16, the classical unitary subgroup of the modular group algebra determines the group.
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Isomorphism problem of Unitary Subgroups of Group Algebras
For finite abelian p-groups, for 2-groups of maximal class, and for 2-groups of order at most 16, the classical unitary subgroup of the modular group algebra determines the group.