The notion of a crossed module, originally formulated by J. H. C. Whitehead within the framework of combinatorial homotopy theory, has emerged as a fundamental structure in modern algebra and topology. Over the decades, crossed modules have been recognized as powerful tools with broad applicability across diverse mathematical disciplines, including category theory, group homology and cohomology, homotopy theory, abstract algebra, and K-theory. They serve as a unifying concept that bridges algebraic and topological methods, offering a rich framework for both theoretical exploration and practical computation. Subsequent research has significantly expanded upon Whitehead’s foundational work. For instance, Alp provided a detailed exposition of the actor crossed module in the context of algebroids, shedding light on its structural properties and categorical significance. Furthermore, Dehghanizadeh and Davvaz have systematically investigated several specialized classes of crossed modules, such as nilpotent, solvable, and n-complete crossed modules, as well as their representation theory, thereby extending the scope of applications and deepening our understanding of these algebraic objects. In the present paper, we continue this line of inquiry by introducing and studying novel concepts related to the center of a crossed module, the group of central automorphisms, and the family of n-central automorphism groups. Our analysis not only formalizes these notions within a rigorous categorical framework but also establishes several new results that clarify the interplay between centrality conditions and the intrinsic algebraic structure of crossed modules. These contributions aim to enrich the current theory and pave the way for future developments in the intersection of algebra, topology, and category theory.
Alp et al. (Wed,) studied this question.