A Specht-to-permutation homomorphism can violate dominance in characteristic two

Over ${\mathbb{F}}_2$, the Specht module $S^{(1,1)}$ for $S_2$ is generated by the nonzero polytabloid $\{1\mid 2\}+\{2\mid 1\}$, which the transposition fixes. It is therefore a one-dimensional trivial module, as is the Young permutation module $M^{(2)}$. There is a nonzero equivariant map $S^{(1,1)}\to M^{(2)}$, although $(1,1)$ does not dominate $(2)$. The familiar dominance implication for a map from an entire permutation module cannot be transferred automatically to a map defined only on a Specht submodule: in modular characteristic an equivariant extension to the larger module need not exist.