Let $p$ be a prime, and consider $S_{p}$, the symmetric group on $p$ elements, thought of here as the group of bijections of $\mathbb{Z}/p\mathbb{Z}$. It is well known that, given a cyclic permutation $\sigma$ whose orbit contains all the elements of $\mathbb{Z}/p\mathbb{Z}$, the primality of $p$ guarantees that, for any transposition $\tau$, the elements $\sigma$ and $\tau$ then generate the entirety of $S_{p}$.
Now, let $\iota$ be an involution of $\mathbb{Z}/p\mathbb{Z}$ (i.e., a non-identity element of $S_{p}$ of order 2). Do $\sigma$ and $\iota$ then necessarily generate $S_{p}$? If not, are there any conditions on $\iota$ more general than "$\iota$ is a transposition" which will guarantee that $\sigma$ and $\iota$ generate $S_{p}$?