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In this note we generalize the extended Brauer quotient defined by L. Puig and Y. Zhou in [4, Section 3] to the case of N-interior G-algebras, where N is a normal subgroup of a finite group G. We use this extended Brauer quotient on permutation algebras to establish some correspondences that generalize known results that hold in the group algebra case.

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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 Staic's construction of symmetric group cohomology. We give explicit embeddings and connecting homomorphisms between the symmetric cohomology spaces and symmetric Hochschild cohomology of twisted group algebras.

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