On transitive permutation groups with bounded movement

Let $G$ be a permutation group on a set $\Omega$. Then for each $g\in G$, we define the movement of $g$, denoted by ${\rm move}(g)$, the maximal cardinality $|\Delta^{g}\backslash \Delta|$ of $\Delta^{g}\backslash \Delta$ over all subsets $\Delta$ of $\Omega$. And the movement of $G$ is defined as the maximum of ${\rm move}(g)$ over all $g\in G$, denoted by ${\rm move}(G)$. A permutation group $G$ is said to have bounded movement if it has movement bounded by some positive integer $m$, that is ${\rm move}(G)\leq m$. In this paper, we consider the finite transitive permutation groups $G$ with movement ${\rm move}(G)=m$ for some positive integer $m>4$, where $G$ is not a $2$-group but in which every non-identity element has the movement $m$ or $m-4$, and there is at least one non-identity element that has the movement $m-4$. We give a characterization for elements of $G$ in Theorem 1.1. Further, we apply Theorem 1.1 to characterize transitive permutation group $G$ in Theorem 1.2. These results give a partial answer to the open problem posed by the authors in 2024.