I prefer to proceed with a concrete example if I may. I appreciate that the answer might well be better explained with group theory, geometry and/or notions from probability theory, which I welcome.

Let $\{x\}$ denote the fractional part of $x$, $(p,q)=1$ and $p\leq q$. If $n\in\mathbb{Z}_q$, then the set of $\phi(q)$ functions defined by

$$f_p(n)=q\{pn/q\}$$

form the complete set of automorphisms of $\mathbb{Z}_q$.

As we vary $p$ through $\mathbb{Z}_q^{\times}$, it is not apparent to me that there is prescribed order in which the elements of $\mathbb{Z}_q$ reappear, neither do I see a rule not involving the function $\{x\}$.

Say we choose some large $q$, then is it the case that order in which the elements reappear exhibits some sense of randomness, and in what sense if so?