  In the theory of crossed modules, considering arbitrary self-actions instead
of conjugation allows for the extension of the concept of crossed modules and
thus the notion of generalized crossed module emerges. In this paper we give a
precise definition for generalized cat$^1$-groups and obtain a functor from the
category of generalized cat$^1$-groups to generalized crossed modules. Further,
we introduce the notions of coverings and liftings for generalized crossed
modules and investigate properties of these structures. Main objective of this
study is to obtain an equivalence between the category of coverings and the
category of liftings of a given generalized crossed module $(A,B,\alpha)$.
