The isomorphism problem of unitary subgroups of modular group algebras

Let V∗(FG) be the normalized unitary subgroup of the modular group algebra FG of a finite p-group G over a finite field F with the classical involution ∗. We investigate the isomorphism problem for the group V∗(FG), i.e., we pose the question when the group algebra FG is uniquely determined by V∗(FG). We give affirmative answers for classes of finite abelian p-groups, 2-groups of maximal class and non-abelian 2-groups of order at most 16.