On the twisted group ring isomorphism problem for a class of groups

The twisted group ring isomorphism problem (TGRIP) is a variation of the classical group ring isomorphism problem. It asks whether the ring structure of the twisted group ring determines the group up to isomorphism. In this article, we study the TGRIP for direct product and central product of groups. We provide some criteria to answer the TGRIP for groups by answering the TGRIP for certain associated quotients. As an application of these results, we provide several examples. Finally, we answer the TGRIP for extra-special p-groups, and for all but five groups of order $p^5$, where $p\ge 5$ is prime.