Let \(Q\) be a finite group acting on a group \(G\) as a group automorphisms, \(C_\ast(G)\) the bar complex, \(H^Q_\ast(G,A)\) the homology of invariant group chains and \(H^\ast_Q(G,A)\) the cohomology invariant, both defined by K. P. Knudson [J. Algebra 298, No. 1, 15–33 (2006; Zbl 1107.20041)].
The authors define the Tate homology of invariants \(\hat{H}^Q_\ast(G,A)\) and the Tate cohomology of invariants \(\hat{H}^\ast_Q(G,A)\). When the coefficient \(A=\mathbb{Z}\), the group of the integers, it is shown an isomorphim \(\hat{H}^i_Q(G,\mathbb{Z})\cong\hat{H}^Q_{i-1}(G, \mathbb{Z})\). Furthermore, the homology and cohomology of invariant group chains are dual, \(H^i_Q(G,\mathbb{Z})\cong H^Q_{i-1}(G, \mathbb{Z})\), \(i\ge 2\). At the end, the authors apply Invariant Universal Coefficient Theorem to calculate the (co)homology of invariant group chains for \(Q =\mathbb{Z}_2\), \(G = \mathbb{Z}_{2^s}\) and \(A = \mathbb{Z}_m\).