When is the action of $G$ on $\text{Syl}_p(G)$ by conjugation is double transitive?

Question

We know that the action of $G$ on $\text{Syl}_p(G)$ by conjugation is transitive. I wonder when this action can be double transitive on $\text{Syl}_p(G)$.

Thanks for your help.

Answer 1

If $G$ acts doubly transitively on its Sylow $p$-subgroups, and $K$ is the subgroup that normalizes all Sylow $p$-subgroups, then $G/K$ is isomorphic to a doubly transitive group in which the point stabilizer has a normal Sylow $p$-subgroup.

Finite doubly transitive groups have been classified. The examples with this property include ${\rm PSL}(2,q)$, ${\rm PSU}(3,q)$, ${\rm Sz}(2^{2n+1})$ and $R(3^{2n+1})$ (Ree groups). There are also some affine examples - i.e. goups with a regular normal elementary abelian subgroup . For example the groups ${\rm AGL}(1,q)$ have this property for primes dividing $q-1$.