Let $G$ be a nonregular, transitive permutation group acting on $\Omega$ such that each nontrivial element either fixes no point or exactly $p$ points for some prime $p$. Further suppose that for $g \notin N_G(G_{\alpha})$ we have $$ G_{\alpha} \cap G_{\alpha}^g = 1. $$
If $p^2 ~ | ~ |\Omega|$ and $p ~ | ~ |G_{\alpha}|$, does this imply that the Sylow $p$-subgroups of $G$ have maximal class?
A $p$-group of order $p^{k+1}$ has maximal class, if its nilpotency class is $k$, see for example here.