Proving isomorphism $H^r_T(G,M)\rightarrow H_T^{r+2}(G,M) $ for cyclic group using cup products

Question

Suppose that we have cyclic group $G$ and a $G$-module $M$. Let $ \chi $ be isomorphism $G \rightarrow \mathbb Z /n \mathbb Z $ sending generator $\sigma \in G$ to $1+n \mathbb Z $. Consider $$0 \rightarrow \mathbb Z \xrightarrow{n} \mathbb Z \rightarrow \mathbb Z /n \mathbb Z \rightarrow 0$$ From this we obtain map $\delta : H^1(G, \mathbb Z /n \mathbb Z )\rightarrow H^2(G, \mathbb Z ) $.
I wish to prove that the map $H^r_T(G,M)\rightarrow H_T^{r+2}(G,M) $ (subscript $T$ stands for Tate cohomology group) defined by $b \mapsto \delta_{\chi} \smile b $ is isomorphism.
I know there is a alternate proof for this isomorphism but I want to prove specifically using cup products.
I do not know where to start. I tried to prove isomorphism for cocycles. Another idea I had was to prove for $r=0$ and then somehow use dimension shifting. But I don't see how to pull that proof.
Any help is appreciated and feel free to give any reference.