= A Specht-to-permutation homomorphism can violate dominance in characteristic two
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Over $\mathbb F_2$, the <Specht module> $S^{(1,1)}$ for $S_2$ is generated by the nonzero polytabloid $\{1\mid2\}+\{2\mid1\}$, 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.
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