I didn't get the argument of Example 7.3.11, page 123, from the representation theory book of Peter Webb, available also online at:
In this example it is shown that the trivial module $\mathbb Z$ does not have a projective cover over $ZG$-mod where $G$ is the cyclic group with two elements.
I have understood the first part where it is shown that the augmentation map $\epsilon : \mathbb ZG \rightarrow \mathbb Z$ is not an essential epimorfism (i.e., the kernel is not superfluous.)
Then indeed, it follows from a previous proposition from the same section that if it were a projective cover for $\mathbb Z$ then that projective cover should be a summand of the regular module $\mathbb ZG$. Thus one can write $$\mathbb ZG =P\oplus Q$$ and $P \rightarrow \mathbb Z$ surjects onto $\mathbb Z$ by the augmentation map.
My question is what is exactly is meant at the end by reduction modulo $2$? Indeed one has a surjective algebra homomorphism $$\pi: \mathbb ZG \rightarrow \mathbb F_2G$$ which gives a decomposition of left ideals
$$\mathbb F_2G=\pi (P) + \pi(Q)$$ I also know that $\mathbb F_2G$ is an indecomposable module over itself. But in order to get a contradiction one has to prove that the above sum is direct. The question I have is why the above sum should be a direct sum and why both terms $\pi(P)$ and $\pi(Q)$ are not zero?