$\DeclareMathOperator{\Res}{Res} \DeclareMathOperator{\Cor}{Cor}$ Let $H$ be a subgroup of index $n$ of a finite group $G$, and let $M$ be a $G$-module, that is, an abelian group on which $G$ acts. Consider the restriction and corestriction homomorphisms in Tate cohomology: \begin{align*} \Res\colon\, &H^{-1}(G,M) \to H^{-1}(H,M),\\ \Cor\colon\, &H^{-1}(H,M) \to H^{-1}(G,M). \end{align*} Since $$ \Cor \circ \Res =n,$$ we know that for $\xi \in H^{-1}(G, M)$, $$\text{if}\ \, \Res \xi=0,\ \text{then}\ \, n\xi=0.$$
Question: What is an example of $(G,H,M,\xi)$ such that $$n\xi=0,\ \ \text{but}\ \ \Res \xi \neq 0\ ?$$