Augmentation subrepresentation of a permutation representation

For a finite $G$-set $X$, the coefficient-sum map $\epsilon :\mathbb{C}[X]\to \mathbb{C}$ is equivariant. Its kernelis the augmentation subrepresentation. If $X$ is nonempty, then

For a transitive finite $G$-set $X$ with at least two points, the augmentation subrepresentation of a permutation representation is irreducible if and only if the action is two-transitive. Indeed, writing the permutation character as $\pi =1+{\sum }_i{m}_i{\chi }_i$ givesThe augmentation subrepresentation is irreducible exactly when this norm is two. By the character norm of a permutation representation, that means that the diagonal and the off-diagonal are the only two orbits on $X\times X$, which is exactly two-transitivity.