  In this paper we define the notions of normal subcrossed module and quotient
crossed module within groups with operations; and using the equivalence of
crossed modules over groups with operations and internal groupoids we prove how
normality and quotient concepts are related in these two categories. Further we
prove an equivalence of crossed modules over groups with operations and
cat$^1$-groups with operations for a certain algebraic category; and then by
this equivalence we determine normal and quotient objects in the category of
cat$^{1}$-groups with operations. Finally we characterize the coverings of
cat$^{1}$-groups with operations.
