induced sequence of an exact sequence of G-modules

Question

Let $G$ be a group. Here the abelian group $\mathbb{Z}$ is assumed to have trivial $G$-action. A short exact sequence $$0\longrightarrow A\longrightarrow B\longrightarrow C\longrightarrow 0$$ of $G$-modules induces an exact sequence $$0\longrightarrow A^{G}\longrightarrow B^{G}\longrightarrow C^{G}$$ and this sequence cannot be extended to a short exact sequence in general. I tried to find a concrete example of short exact sequence of $G$-modules such that the induced sequence is not a short exact sequence. It is known that for any $G$-module $A$, $A^{G}\cong \text{Hom}_{\mathbb{Z}[G]}(\mathbb{Z},A)$. So I replaced my problem by the problem finding a group $G$ such that $\mathbb{Z}$ as a $\mathbb{Z}[G]$-module (the action of $\mathbb{Z}[G]$ on $\mathbb{Z}$ is induced from the trivial action of $G$ on $\mathbb{Z}$) is not projective. But I couldn't. If someone know an example, please let me know.

Answer 1

Let $G=\Bbb{Z}$. Then $\Bbb{Z}[G]=\Bbb{Z}[t,t^{-1}]$. If $\Bbb{Z}$ with the trivial action was projective, it would be a direct summand of a free module. This would imply that some free $\Bbb{Z}[t,t^{-1}]$-module has non-trivial fixed points, which is wrong. Thus $\Bbb{Z}$ is not a projective $\Bbb{Z}[t,t^{-1}]$-module.