Free crossed modules A crossed module (over groups) $\mathcal{M} = (H,G,\partial)$ is a homomorphism $\partial\colon H \to G$ (called the boundary) together with an action $\alpha\colon (g,h) \mapsto {}^gh$ of $G$ on $H$ such that the following two axioms are satisfied:


*

*$\partial({}^gh) = g\partial(h)g^{-1}$

*${}^{\partial(h)}h' = hh'h^{-1}$


for all $h,h' \in H$ and $g \in G$.
Given two crossed modules $\mathcal{M} = (H,G,\partial)$ and $\mathcal{N} = (H',G',\partial')$, a morphism $$(\mu,\nu)\colon \mathcal{M} \to \mathcal{N}$$ is a pair of group homomorphisms such that $\mu$ and $\nu$ interacts well with the boundaries of $\mathcal{M}$ and $\mathcal{N}$. Moreover, the action is preserved, in the sense that $$\mu({}^gh) = {}^{\nu(g)}\mu(h)$$ for all $h \in H$ and $g \in G$.
With the objects and morphisms defined above, we form the category $\mathsf{CrossedMod}$ of crossed modules.
In Crossed modules and homology [p.44, 1], the authors defined a free functor $$F\colon (\mathsf{Set} \downarrow U) \to \mathsf{CrossedMod},$$ where $U\colon \mathsf{Grp} \to \mathsf{Set}$ is the forgetful functor and $(\mathsf{Set} \downarrow U)$ is the comma category of objects $U$-under $\mathsf{Set}$. They have called $F(f\colon X \to U(G))$ the free crossed module over $f\colon X \to U(G)$.
I am not totally convinced that this indeed forms a crossed module. Are there any other free functors that can be constructed to the category of crossed modules?
I am interested to see how these free crossed modules are useful and specifically where they are useful.
[1] http://www.sciencedirect.com/science/journal/00224049/95/1
 A: I found a partial answer to my question a while ago and so I thought I should put it here. I shall construct free crossed modules that are associated with group presentations. 
The references that I have used to understand this are:


*

*H. J. Baues, Combinatorial homotopy 4-dimensional complexes, Walter de Gruyter Expos. Math. 2 (1991), 94,95.

*A. J. Sieradski, Algebraic topology for two-dimensional complexes, Two-dimensional homotopy and combinatorial group theory 197 (1993), 51–96, London Mathematical Society Lecture Note Series, Cambridge University Press.
Let $X$ be a set and let $F[X]$ be the free group on the set $X$. Let $R
\subseteq F[X]$ be a subset of $F[X]$. The elements of $R$ are called
relators. Then, $\mathcal{P} = \left< X\;|\;R \right>$ is a
presentation of a group. Now, let $E(\mathcal{P})$ denote the free group on the
set $F[X] \times R$ of ordered pairs $(w,r)$, where $w \in F[X]$ and $r \in
R$.
There is a group action $$F[X] \times E(\mathcal{P}) \to E(\mathcal{P})$$
given by $$(v, (w,r)) \mapsto {}^v(w,r) = (vw,r)$$ with $v,w \in F[X]$, $r \in R$ and a group homomorphism $\delta\colon E(\mathcal{P}) \to F[X]$ defined on the basis elements by $\delta(w,r) = wrw^{-1}$.
Notice that $\mathrm{im}(\delta) = N(R)$, the normal closure of $R$ in $F[X]$. The subgroup $I(\mathcal{P}) = \mathrm{ker}(\delta\colon E(\mathcal{P}) \to F[X])$ of $E(\mathcal{P})$ is called the group of identities for the presentation $\mathcal{P}$.
It is easy to see that the homomorphism $\delta\colon E(\mathcal{P}) \to F[X]$
is a pre-crossed module (just the first axiom is satisfied). That is, for all $v,w \in E(\mathcal{P})$ and $r \in R$, we have:
\begin{align*}
  \delta({}^v(w,r)) &= \delta((vw,r))\\
                   &= vwr(vw)^{-1}, \text{ by defintion of $\delta$}\\
                   &= v\delta(w,r)v^{-1}.
\end{align*}
The action $F[X] \times E(\mathcal{P}) \to E(\mathcal{P})$ of $\delta(w,r) =
wrw^{-1}$ in $F[X]$ on $(v,s)$ in $E(\mathcal{P})$ gives us ${}^{(wrw^{-1})}(v,s)
=(wrw^{-1}v,s)$ but we notice that although $(wrw^{-1}v,s)$ does not equal the
conjugate $(w,r)(v,s)(w,r)^{-1}$ of $(v,s)$ by $(w,r)$ in $E(\mathcal{P})$, they
have the same boundary in $F[X]$:
\begin{align*}
  \delta((wrw^{-1}v,s)) &= (wrw^{-1}v)s(wrw^{-1}v)^{-1}\\
                        &= \delta((w,r)(v,s)(w,r)^{-1}).
\end{align*}
And thus, this implies that $$(w,r)(v,s)(w,r)^{-1}(wrw^{-1}v,s)^{-1}$$ in $E(\mathcal{P})$ is in kernel of $\delta$.
Elements in $E(\mathcal{P})$ of the form
$$(w,r)(v,s)(w,r)^{-1}\left(^{\delta((w,r))}(v,s)\right)^{-1}$$ with $(w,r),(v,s) \in E(\mathcal{P})$ are called Peiffer commutators.
It is obvious that the Peiffer commutators lie in $\mathrm{ker}(\delta) =
I(\mathcal{P})$. Hence, the normal closure $P(\mathcal{P}) \subseteq
I(\mathcal{P})$ in $E(\mathcal{P})$ is called the Peiffer group for the
presentation $\mathcal{P}$.
The action $F[X] \times E(\mathcal{P}) \to E(\mathcal{P})$ and homomorphism
$\delta$ induce an action of $F[X]$ on the quotient group $C(\mathcal{P}) =
E(\mathcal{P})/P(\mathcal{P})$
$$F[X] \times C(\mathcal{P}) \to C(\mathcal{P})$$
and a boundary homomorphism
$$\partial\colon C(\mathcal{P}) \to F[X]$$
given on the generators by $\partial((w,r)P(\mathcal{P})) = wrw^{-1}$.
Therefore, $\mathscr{F} = (C(\mathcal{P}), F[X], \partial)$ constitute a free
crossed module associated with the group presentation $\mathcal{P}$. Finally, we see that $\mathrm{ker}(\partial) = I(\mathcal{P})/P(\mathcal{P})$ and $\mathrm{im}(\partial) = N(R)$.
A: You say:
I am interested to see how these free crossed modules are useful and specifically where they are useful.
There are two main areas and these correspond to the two sources you mention. The two areas are combinatorial group theory and homotopy theory.
Perhaps a better source for the first of these is the paper by Brown and Huebschmann on Identities amongst Relations.  This can then be followed up by papers by Loday (Homotopical Syzygies,) and a lovely paper by Kapranov and Saito (Hidden Stasheff polytopes in algebraic K- theory and in the space of Morse functions).  
For the homotopical links, have a look at the big book by Brown, Higgins and Sivera.  (Internet searches will find you fuller details. I have some notes (https://ncatlab.org/timporter/show/crossed+menagerie) that may help. I realise it is a long time since you asked your question but hopefully this is still useful and may be of use to others as well.)
A: More information, as Tim Porter says., is in the book Nonabelian Algebraic Topology, and a later description of the background is in this paper on Modelling and Computing Homotopy Types: I. 
