We generalize quantum Drinfeld Hecke algebras by incorporating a
-cocycle
on the associated finite group. We identify these algebras as specializations of
deformations of twisted skew group algebras, giving an explicit connection to
Hochschild cohomology. We classify these algebras for diagonal actions, as well as for
the symmetric groups with their natural representations. Our results show that the
parameter spaces for the symmetric groups in the twisted setting is smaller than in
the untwisted setting.
