  We show that there is an action of the symmetric group on the Hochschild
cochain complex of a twisted group algebra with coefficients in a bimodule.
This allows us to define the symmetric Hochschild cohomology of twisted group
algebras, similarly to th construction of symmetric group cohomology due to
Staic. We give explicit embeddings and connecting homomorphisms between the
symmetric cohomology spaces and symmetric Hochschild cohomology of twisted
group algebras.
