Subnormal closure of a homomorphism

Let $\phi \text{\hspace{0.17em}⁣}:\mathrm{\Gamma }\to G$ 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 $\mathrm{\Gamma }\stackrel{\psi }{\to }M\stackrel{n}{\to }G$ of $\phi ,$ with $n$ a subnormal map. We search for a universal such factorization. When $\mathrm{\Gamma }$ and $G$ are finite we show that such universal factorization exists: $\mathrm{\Gamma }\to {\mathrm{\Gamma }}_{\mathrm{\infty }}\to G,$ where ${\mathrm{\Gamma }}_{\mathrm{\infty }}$ is a hypercentral extension of the subnormal closure $\mathcal{C}$ of $\phi (\mathrm{\Gamma })$ in $G$ (i.e.~the kernel of the extension ${\mathrm{\Gamma }}_{\mathrm{\infty }}\to \mathcal{C}$ is contained in the hypercenter of ${\mathrm{\Gamma }}_{\mathrm{\infty }}$). This is closely related to the a relative version of the Bousfield-Kan $\mathbb{Z}$-completion tower of a space. The group ${\mathrm{\Gamma }}_{\mathrm{\infty }}$ is the inverse limit of the normal closures tower of $\phi$ introduced by us in a recent paper. We prove several stability and finiteness properties of the tower and its inverse limit ${\mathrm{\Gamma }}_{\mathrm{\infty }}$.