Given a prime $p$, find a formal power series $f$ in the ring $F_p[[X]]$ such that $f^{\circ p^2}(X) = X$, where $\circ$ denotes composition and $f^{\circ n} = \underbrace{f\circ f\circ \cdots\circ f}_{n\text{ times}}$.
If possible, find power series of other orders not equal to $1$ or $p$.