Exercise 6, § 1

Let $E$ be a set with 6 elements and $F$ the set of structures of the projective line over the field $\mathbf{F}_5$ on $E$. Denote by $S_E$ the group of permutations of $E$; for any $\sigma \in S_E$ denote by $\tilde{\sigma}$ the permutation of $F$ induced by $\sigma$ by transport of structure. Show that there exists a bijection $u$ from $E$ to $F$, and that the map $\sigma \mapsto u^{-1} \tilde{\sigma} u$ is an outer automorphism of $S_E$ (if $s$ is a transposition, $u^{-1} \tilde{s} u$ has three orbits of two elements).