TITLE: Projectives in the category of discrete G-modules 
QUESTION [5 upvotes]: If $G$ is a profinite group, then the category $Mod(G)$ of discrete $G$-modules has sufficiently many injectives (Neukirch, Schmidt, Wingberg: Cohomology of Number Fields, 2.6.5). 
Since the cited book says nothing on projectives within this category, I guess $Mod(G)$ doesn't have sufficiently many projectives. Can you give me an example of a discrete $G$-module $M$ such that there is no epimorphism  $P \to M$ with $P$ projective ?  If there are little projectives, is it even possible to classify them ? 
Added: Note that a discrete $G$-module $M$ is an abelian group (with the discrete topology) with a continuous $G$-action $G \times M \to M$ (N-S-W, 1.1.5). For example $\mathbb{Z}$ with trivial $G$-action is a discrete $G$-module.

REPLY [2 votes]: Here is a link a proof that the category $\text{Mod}(G)$ does not have enough projectives, when $G$ is an infinite profinite group. I think the proof is fairly elementary. There might be a few details missing though - sorry about that.