Find an example about cohomologically trivial

Question

Give an example of a $G$-module $M$, such that $\hat{H}^{*}(G,M)=0$, but $M$ is not cohomologically trivial. Here $\hat{H}^{*}(G,-)$ means Tate Cohomology.

Answer 1

This hint is from Serre's Local fields:

Take $G$ to be the cyclic group of order 6 and let $A = \mathbb{Z}/3\mathbb{Z}$. Let $G$ operate on $A$ by $x \mapsto -x$. Then show that $\hat{H^{0}}(H,A) \neq 0$, where $H$ is the subgroup of order 3.

Hope that helps.