Arithmetic functions and fixed points of powers of permutations

Let $\sigma$ be a permutation of a nonempty finite or countably infinite set $X$ and let $F_X\left( \sigma^k\right)$ count the number of fixed points of the $k$th power of $\sigma$. This paper explains how the arithmetic function $k \mapsto \left(F_X\left( \sigma^k\right) \right)_{k=1}^{\infty}$ determines the conjugacy class of the permutation $\sigma$, constructs an algorithm to compute the conjugacy class from the fixed point counting function $F_X\left( \sigma^k\right)$, and describes the arithmetic functions that are fixed point counting functions of permutations.