  Similarly to how the classical group ring isomorphism problem asks, for a
commutative ring $R$, which information about a finite group $G$ is encoded in
the group ring $RG$, the twisted group ring isomorphism problem asks which
information about $G$ is encoded in all the twisted group rings of $G$ over
$R$.
  We investigate this problem over fields. We start with abelian groups and
show how the results depend on the roots of unity in $R$. In order to deal with
non-abelian groups we construct a generalization of a Schur cover which exists
also when $R$ is not an algebraically closed field, but still linearizes all
projective representations of a group. We then show that groups from the
celebrated example of Everett Dade which have isomorphic group algebras over
any field can be distinguished by their twisted group algebras over finite
fields.
