Permutable subgroup is normalized by any trivially intersecting infinite cyclic group

Statement
Suppose $$H$$ is a fact about::permutable subgroup of a group $$G$$ and $$K$$ is an infinite cyclic subgroup of $$G$$ that intersects $$H$$ trivially. Then, $$K$$ normalizes $$H$$: in other words, $$K$$ is contained in the normalizer $$N_G(H)$$.