The indecomposable projective $\mathbb{F}_pG$-module with $UJ/J\cong \mathbb{F}_p$

Question

Let:

1. $G$ be a finite group;
2. $p$ be prime;
3. $J$ be the Jacobson radical of $\mathbb{F}_pG$.

A paper I'm trying to read mentions the following object:

The indecomposable projective $\mathbb{F}_pG$-module $U$ with $U/UJ\cong\mathbb{F}_p$

It is then also claimed that $U$ is a direct summand of $\mathbb{F}_pG$.

1. Why does this object exist?
2. Why is it unique?
3. Why is it a direct summand of $\mathbb{F}_pG$.

I know all the definitions of the terms mentioned, but not experienced with some of them.

The paper is "The Presentation Rank of a Direct Product of Finite Groups" / Cossey, Gruenberg, Kovacs (Journal Of Alegebra 28, 597-603 (1974)).

EDIT: If relevant, it might be understood from context that $p$ divides $|G|$, but I'm nore sure.