  We continue our investigation of a variation of the group ring isomorphism
problem for twisted group algebras. Contrary to previous work, we include
cohomology classes which do not contain any cocycle of finite order. This
allows us to study the problem in particular over any field of characteristic
$0$. We prove that there are finite groups $G$ and $H$ which can not be
distinguished by their rational twisted group algebras, while $G$ and $H$ can
be identified by their semi-simple twisted group algebras over other fields.
This is in contrast with the fact that the structural information on $G$ which
can obtained from all the semi-simple group algebras of $G$ is already encoded
in its rational group algebra.
  We further show that for an odd prime $p$ there are groups of order $p^4$
which can not be distinguished by their twisted group algebras over $F$ for any
field $F$ of characteristic different from $p$. On the other hand we prove that
the groups constructed by E. Dade, which have isomorphic group algebras over
any field, can be distinguished by their rational twisted group algebras. We
also answer a question about sufficient conditions for the twisted group ring
isomorphism problem to hold over the complex numbers.
