R\'edei permutations with cycles of the same length

Let $\mathbb{F}_q$ be a finite field of odd characteristic. We study R\'edei functions that induce permutations over $\mathbb{P}^1(\mathbb{F}_q)$ whose cycle decomposition contains only cycles of length $1$ and $j$, for an integer $j\geq 2$. When $j$ is $4$ or a prime number, we give necessary and sufficient conditions for a R\'edei permutation of this type to exist over $\mathbb{P}^1(\mathbb{F}_q)$, characterize R\'edei permutations consisting of $1$- and $j$-cycles, and determine their total number. We also present explicit formulas for R\'edei involutions based on the number of fixed points, and procedures to construct R\'edei permutations with a prescribed number of fixed points and $j$-cycles for $j \in \{3,4,5\}$.