Normal divisors of a 2-transitive group of automorphisms of a linearly ordered set

The main result of the paper is a description of the normal structure of the groups $\operatorname{Aut}(X,\leqslant)$, where $X$ is a linearly ordered set satisfying one of the following equivalent conditions: I. $\operatorname{Aut}(X,\leqslant)$ is 2-transitive. II. $\operatorname{Aut}(X,\leqslant)$ is $k$-transitive. III. $X$ does not have a greatest or a least element, and any two intervals $[a,b]$, $a<b$ and $[c,d]$, $c<d$, are similar. IV. $\operatorname{Aut}(X,\leqslant)$ is a 0-primitive, transitive, nonregular permutation group.
Main theorem. {\it Suppose $\operatorname{Aut}(X,\leqslant)$ is $2$-transitive. Then $\overline A,$ $\overset\rightarrow A$ and $\overset\leftarrow A$ are the only nontrivial normal and subnormal subgroups of $\operatorname{Aut}(X,\leqslant)$. Here
\begin{gather*} \overset\leftarrow A=\{g\in\operatorname{Aut}(X,\leqslant)\mid \operatorname{Tr}g\text{ is bounded below}\},\\ \overset\rightarrow A=\{g\in\operatorname{Aut}(X,\leqslant)\mid \operatorname{Tr}g\text{ is bounded above}\},\\ \overline A=\overset\rightarrow A\cap\overset\leftarrow A,\qquad\operatorname{Tr}g=\{x\in X\mid g(x)\ne x\}. \end{gather*}
}
Bibliography: 21 titles.