  Let $\varphi\colon\Gamma\to G$ be a homomorphism of groups. We consider
factorizations $\Gamma\xrightarrow{f} M\xrightarrow{g} G$ of $\varphi$ such
that either $g$ or $f$ are universal normal maps (namely, crossed modules).
These two factorizations are natural generalizations of the usual normal
closure and normalizer of a subgroup. Iterating these universal factorizations
yield towers related, on the one hand, to hypercentral group extensions,
Bousfield's localizations, and relative Schur multipliers. Dually, they extend
to a relative situation, the automorphisms tower of a centerless group. Our
constructions have strong ties to topological constructions.
