$OD$-Characterization of the Automorphism Groups of Simple $K_{3}$-Groups

The degree pattern of a finite group $G$ associated to its prime graph has been introduced in \cite{y7} and denoted by $D(G)$. The group $G$ is called $k$-fold $OD$ characteri zable if there exist exactly $k$ non-isomorphic groups $H$ satisfying conditions (1) $|G|=|H|$ and (2) $D(G)=D(H)$. Moreover, a $1$-fold $OD$-characterizable group is simply called $OD$-characterizable group. In this problem, those groups with connected prime graphs are somewhat much difficult to be solved. In the present paper, we continue to this investigation and show that the automorphism groups of simple $K_{3}$-groups are characterized by their orders and degree patterns. In fact, the automorphism groups of simple $K_{3}$-groups except $A_{6}$ and $U_{4}(2)$ are $OD$-characterizable. Moreover, $Aut(A_{6})$ is $4$-fold $OD$-characterizable and $Aut(U_{4}(2))$ is at least $4$-fold $OD$-characterizable.