The aim of this work is to offer a definition of the sharp partial order for the set of Core-Nilpotent endomorphisms. The main properties of this partial order are studied, generalizing the notion of sharp partial order of finite square matrices. In order to do so, the structure of the set of {1}-commuting inverses of a Core-Nilpotent endomorphism is described, and in particular, the structure of the set of {1}-commuting inverses of finite square matrices with entries in an arbitrary field is determined and an algorithm for its computation is provided.