I want to construct an example of Herbrand quotient's hexagon diagram.
Let $0 \to p\mathbb{Z} \to \mathbb{Z} \to \mathbb{Z}/p\mathbb{Z} \to 0$ be a short exact sequence of $\mathbb{Z}$-modules and we get induced long exact sequence
$$H^0(\mathbb{Z}, p\mathbb{Z}) \to H^0(\mathbb{Z}, \mathbb{Z}) \to H^0(\mathbb{Z}, \mathbb{Z}/p\mathbb{Z}) \to H^1(\mathbb{Z}, p\mathbb{Z}) \to H^1(\mathbb{Z}, \mathbb{Z}) \to$$ $$H^1(\mathbb{Z}, \mathbb{Z}/p\mathbb{Z}) \to H^2(\mathbb{Z}, p\mathbb{Z})\to \dots $$
and $H^0(\mathbb{Z}, p\mathbb{Z}) \cong H^2(\mathbb{Z}, \mathbb{Z}/p\mathbb{Z})$, so we can get Herbrand's hexagon diagram of $\mathbb{Z}$-modules.
$$H^0(\mathbb{Z}, p\mathbb{Z}) \to H^0(\mathbb{Z}, \mathbb{Z}) \to H^0(\mathbb{Z}, \mathbb{Z}/p\mathbb{Z}) \to H^1(\mathbb{Z}, p\mathbb{Z}) \to H^1(\mathbb{Z}, \mathbb{Z}) \to$$ $$H^1(\mathbb{Z}, \mathbb{Z}/p\mathbb{Z}) \to H^2(\mathbb{Z},p\mathbb{Z})$$
But I have stuck with calculating each component of the sequence above. I would be appreciated if you could give me help completing the exact sequence. If you know more good example of Herbrand quotient hexagon, such an example is also appreciated, thank you.
P.S.
Thanks to a comment, I noticed I cannot make an example of Herbrand quotient hexagon by $\mathbb{Z}$-modules.
I would be appreciated if you could construct example using $\mathbb{Z}/p\mathbb{Z}$-modules.