Invariants of Specht modules

In [14] Hemmer conjectures that the module of fixed points for the symmetric group ΣmΣm of a Specht module for ΣnΣn (with n>mn>m), over a field of positive characteristic p  , has a Specht series, when viewed as a Σn−mΣn−m-module. We provide a counterexample for each prime p  . The examples have the same form for p≥5p≥5 and we treat the cases p=3p=3 and p=2p=2 separately.