On permutation modules and decomposition numbers for symmetric groups

We study the indecomposable summands of the permutation module obtained by inducing the trivial $\mathbb{F}(S_a\wr S_n)$-module to the full symmetric group $S_{an}$ for any field $\mathbb{F}$ of odd prime characteristic $p$ such that $a<p\leq n$. In particular we characterize the vertices of such indecomposable summands. As a corollary we will disprove a modular version of Foulkes' Conjecture. In the second part of the article we will use this information to give a new description of some columns of the decomposition matrices of symmetric groups in terms of the ordinary character of the Foulkes module $\phi^{(a^n)}$.