Weakly normal subgroup of symmetric group must move more than half the elements

Statement
Suppose $$S$$ is a set, $$G = \operatorname{Sym}(S)$$ is the fact about::symmetric group on $$S$$, and $$H$$ is a nontrivial fact about::weakly normal subgroup of $$G$$. Define:

$$\operatorname{supp}(H) = \{ a \in S \mid \ \exists \ \sigma \in H, \sigma(a) \ne a \}$$.

Then, the cardinality of $$\operatorname{supp}(H)$$ should be strictly greater than the cardinality of $$S \setminus \operatorname{supp}(H)$$.

Note that since any fact about::pronormal subgroup and any fact about::paranormal subgroup is weakly normal, we get corresponding results for pronormal and paranormal subgroups. In fact, the proof also shows that the corresponding statement holds for fact about::polynormal subgroups.