Redei permutations with cycles of the same length

Let $mathbb{F}_q$ be a finite field of odd characteristic. We study Redei 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 $jgeq 2$. When $j$ is $4$ or a prime number, we give necessary and sufficient conditions for a Redei permutation of this type to exist over $mathbb{P}^1(mathbb{F}_q)$, characterize Redei permutations consisting of $1$- and $j$-cycles, and determine their total number. We also present explicit formulas for Redei involutions based on the number of fixed points, and procedures to construct Redei permutations with a prescribed number of fixed points and $j$-cycles for $j in {3,4,5}$.