Kaluzhnin--Krasner embedding of precrossed modules

The Kaluzhnin--Krasner embedding establishes that, for a group extension, the middle group can be embedded into the wreath product of its kernel and cokernel. Recently, this construction has been generalised in a categorical framework, recovering both the classical group-theoretic result and its analogue for Lie In this article, we apply this categorical framework to two specific cases: precrossed modules and crossed modules (over groups). For precrossed modules, we introduce a Kaluzhnin--Krasner embedding -- previously unformulated -- by rigorously following the steps of the categorical procedure. In contrast, for crossed modules, we encounter substantial obstacles: while we present partial positive results, we also explain why constructing a full-fledged Kaluzhnin--Krasner embedding in this context is far more difficult.