On groups with the same character degrees as almost simple groups with socle the Mathieu groups

Abstract: Let $G$ be a finite group and $cd(G)$ denote the set of complex irreducible character degrees of $G$. In this paper, we prove that if $G$ is a finite group and $H$ is an almost simple group whose socle is Mathieu group such that $cd(G) =cd(H)$, then there exists an Abelian subgroup $A$ of $G$ such that $G/A$ is isomorphic to $H$. This study is heading towards the study of an extension of Huppert's conjecture (2000) for almost simple groups.