Consider the set $\Pi$ of permutations $\pi: [n] \to [n]$ where each such $\pi$ is an $n$-cycle (i.e., one big cycle consisting of all elements). We know $|\Pi| = (n-1)!$.
I am interested in composing elements of $\Pi$ with an arbitrary permutation $\rho$, i.e., I am looking for some structure of the set $\Pi(\rho)=\{\pi \circ \rho \mid \pi \in \Pi\}$ in terms of some structure of $\rho$. Of course, assuming $n$ is even, the parity of $\rho$ determines the parity of $\pi \circ \rho$. But I do not think this set $\Pi(\rho)$ can be an arbitrary set of $(n-1)!$ permutations with the same parity (for some $\rho$). Is there something known about this (even when assuming some structure of $\rho$)?
An easier question would be to assume $\rho$ as some $k$-cycle ($k \leq n$). In that case, is there something known about the structure of $\Pi(\rho)$?