Subgroup invariant under filtered power automorphism is powered over all primes dividing the denominator

Statement
Suppose $$G$$ is a nilpotent group and $$\sigma$$ is a filtered power automorphism corresponding to a rational number $$r$$. Suppose $$H$$ is a subgroup of $$G$$ satisfying $$\sigma(H) \le H$$. Then, $$H$$ is a powered group for all the primes dividing the denominator of $$r$$ (when written in reduced form).