The isomorphism problem of unitary subgroups of modular group algebras

Let $V_*({F}G)$ be the normalized unitary subgroup of the modular group algebra ${F}G$ of a finite $p$-group $G$ over a finite field ${F}$ with the classical involution $*$. We investigate the isomorphism problem for the group $V_*({F}G)$, i.e., we pose the question when the group algebra ${F}G$ is uniquely determined by $V_*({F}G)$. 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 $16$.