Invariants of Specht Modules

In [14] Hemmer conjectures that the module of fixed points for the symmetric group ${\mathrm{\Sigma }}_m$ of a Specht module for ${\mathrm{\Sigma }}_n$ (with $n>m$), over a field of positive characteristic $p$, has a Specht series, when viewed as a ${\mathrm{\Sigma }}_{n-m}$-module. We provide a counterexample for each prime $p$. The examples have the same form for $p\ge 5$ and we treat the cases $p=3$ and $p=2$ separately.