Does category of finitely generated torsion $G$-modules has enough injectives?

Question

Let $G$ be a profinite group. Then the category of discrete $G$-modules have enough injectives. Now I have a category of finitely generated and torsion $G$-modules with continuous $G$-action. Does injective objects exist in this category. If yes, then how to prove. If no, then any example.

Answer 1

No. For instance, when $G$ is trivial, you just have the category of finite abelian groups, and I claim that there are no nonzero injective objects in this category. Every finite abelian group is a direct sum of cyclic groups of prime power order, so it suffices to show a cyclic group $\mathbb{Z}/(p^n)$ where $p$ is prime and $n>0$ is not injective. To prove this, just consider the extension $$0\to\mathbb{Z}/(p^n)\stackrel{p}\to\mathbb{Z}/(p^{n+1})\to\mathbb{Z}/(p)\to 0$$ which does not split.