Indecomposable summands of Foulkes modules

In this paper we study the modular structure of the permutation module $H^{(2^n)}$ of the symmetric group $S_{2n}$ acting on set partitions of a set of size $2n$ into $n$ sets each of size $2$, defined over a field of odd characteristic $p$. In particular we characterize the vertices of the indecomposable summands of $H^{(2^n)}$ and fully describe all of its indecomposable summands that lie in blocks of $p$-weight at most two. When $2n < 3p$ we show that there is a unique summand of $H^{(2^n)}$ in the principal block of $S_{2n}$ and that this summand exhibits many of the extensions between simple modules in its block.