  Let $G$ be a finite group of order $n$ and let $M$ be a $G$-module. We
construct 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 a
twisting of a $G$-module $M$ defined in Section $5$ and $H^{*}_\lambda(G,M)$ is
a variation of the group cohomology introduced by Zarelua, which in many cases
is isomorphic to the symmetric cohomology of groups defined by Staic. The
groups $H_*^\varkappa(G,M)$ come together with transformations from Tate
cohomology. We find conditions under which these transformations are
isomorphisms.
