  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 the
associated quotients. As an application of these results, we provide several
examples. Finally, we answer the TGRIP for extra-special p-groups, and for the
groups of order $p^5$, where $p \geq 5$ is a prime, except a list of five
groups.
