 Solution

Put $N_G={\sum }_{g\in G}g$. Its image on every module lies in the invariant submodule, since $h{N}_G=N_G$. In the regular module,and this line is the socle of the projective cover $P_k$ of the trivial module.

Suppose $N_{G}M\ne 0$. Choose $m\in M$ with $N_{G}m\ne 0$ and consider the homomorphismIts restriction $f{|}_{P_k}$ is nonzero on $\mathrm{Soc}(P_k)=k{N}_G$. Because $P_k$ is the injective hull of its simple socle, that socle is essential: every nonzero submodule meets it. Hence $\ker (f{|}_{P_k})=0$. The resulting embedding $P_k\hookrightarrow M$ splits because $P_k$ is injective. Since $M$ is indecomposable, $M\cong P_k$.

Conversely, on $P_k$ the image of $N_G$ is its one-dimensional socle. Thus the group norm element detects the trivial projective cover: