  We study finite transitive permutation groups
$G\leqslant\operatorname{Sym}(\Omega)$ such that all orbits of the conjugation
action on $G$ of the normaliser of $G$ in $\operatorname{Sym}(\Omega)$ have
size bounded by some constant. Our results extend recent results, due to the
first author, on finite abstract groups $G$ such that all orbits of the natural
action of the automorphism group $\operatorname{Aut}(G)$ on $G$ have size
bounded by some constant.
