Arithmetic functions and fixed points of powers of permutations

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