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Let $G$ be a finite group of order $n$ and let $M$ be a $G$-module. Weconstruct groups $H_*^\varkappa(G,M)$ for which  $H_k^\varkappa (G,M^{tw}) \cong H^{n-k-1}_\lambda(G,M),$ where $M^{tw}$ is atwisting of a $G$-module $M$ defined in Section $5$ and $H^{*}_\lambda(G,M)$ isa variation of the group cohomology introduced by Zarelua, which in many casesis isomorphic to the symmetric cohomology of groups defined by Staic. Thegroups $H_*^\varkappa(G,M)$ come together with transformations from Tatecohomology. We find conditions under which these transformations areisomorphisms.

Symmetric cohomology of groups and Poincar\'e duality