Subnormal closure of a homomorphism

Let (Formula presented.) be a homomorphism of groups. In this paper we introduce the notion of a subnormal map (the inclusion of a subnormal subgroup into a group being a basic prototype). We then consider factorizations (Formula presented.) of (Formula presented.) with (Formula presented.) a subnormal map, namely a finite composition of the underlying maps of crossed modules. We search for a universal such factorization. When (Formula presented.) and (Formula presented.) are finite we show that such universal factorization exists: (Formula presented.) where (Formula presented.) is a hypercentral extension of the subnormal closure (Formula presented.) of (Formula presented.) in (Formula presented.) (i.e. the kernel of the extension (Formula presented.) is contained in the hypercenter of (Formula presented.)). This is closely related to the a relative version of the Bousfield-Kan (Formula presented.) -completion tower of a space. The group (Formula presented.) is the inverse limit of the normal closures tower of (Formula presented.) introduced by us in a recent paper. We prove several stability and finiteness properties of the tower and its inverse limit (Formula presented.).