Goldman-type theorem for sharply k-transitive permutation groups

Let $G$ be a sharply $k$-transitive permutation group, meaning that $G_{x_1,x_2,\dots,x_{k-1}}$ acts regularly on $X\setminus\{x_1,x_2,\ldots,x_{k-1}\}$. Goldman-type theorem for sharply $k$-transitive groups. A Goldman-type theorem holds for every sharply $k$-transitive permutation group. This is proposed as the first step in an induction strategy for proving a Goldman-type theorem for larger subgroup inclusions arising from $k$-transitive but not $(k+1)$-transitive groups; the statement concerns the sharply transitive cases classified in the surrounding discussion.