Filtered power automorphism

Definition
Suppose $$G$$ is a nilpotent group. Consider a central series of $$G$$ of the form:

$$G = H_1 \ge H_2 \ge H_3 \ge \dots \ge H_n = 1$$

A filtered power automorphism of $$G$$ corresponding to a rational number $$r$$ is an automorphism $$\sigma$$ of $$G$$ such that the following hold:


 * $$\sigma(H_i) = H_i$$ for each $$i$$.
 * Each of the quotient groups $$H_i/H_{i+1}$$ is powered over all primes dividing the numerator or denominator of $$r$$.
 * The induced automorphism by $$\sigma$$ on the quotient group $$H_i/H_{i+1}$$ is the powering map by $$r^i$$.