Centralisers of transitive permutation groups

Question

Let $G$ be a transitive permutation group on a set of size $n$, and suppose $Z(G)=1$ (for instance $G$ is a direct power of a non-abelian simple group). What can we say about the centraliser $K$ of $G$ in $Sym(n)$? I'm interested firstly if there are any restrictions on $K$ independent of degree, and secondly on what role the degree plays.

Answer 1

Of course, there is the classical result that $C_{Sym(n)}(G)$ is a semi-regular subgroup of $Sym(n)$ of cardinality $|Fix(G_{0})|$, where $G_{0}$ is the stabilizer of a point and $Fix(G_{0})$ is the set of points fixed by $G_{0}$.