  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.
