Isomorph-normal implies conjugation-invariantly relatively normal in any ambient group

Statement
Suppose $$H \le K \le G$$ are groups. Suppose, further, that $$H$$ is an fact about::isomorph-normal subgroup of $$K$$: every subgroup of $$K$$ isomorphic to $$H$$ is normal. Then, $$H$$ is a fact about::conjugation-invariantly relatively normal subgroup of $$K$$ relative to $$G$$.