The notion of free crossed module was a major feature of JHC Whitehead's 1949 paper "Combinatorial homotopy II", in which he proved, using methods of transversality and knot theory, that the crossed module $$\pi_2(X \cup \{e^2_\lambda\},X,x) \to \pi_1(X,x) $$ is free on the characteristic maps of the $2$-cells. This theorem is sometimes mentioned but rarely proved in topology texts. Work of Philip Higgins and I showed how this theorem was a special case of a $2$-dim van Kampen type theorem, i.e. a colimit theorem, and the full story of this is given in Part I of the book partially titled Nonabelian Algebraic Topology, EMS 2011 (NAT). Thus the more general theorem determines $$\pi_2(X \cup _f CA,A,x) \to \pi_1(X,x)$$ for $A$ connected, in terms of the induced morphism $f_*: \pi_1(A,a) \to \pi_1(X,x)$, so that Whitehead's theorem is the case $A$ is a wedge of circles.
Feb 14, 2016
This question could also be looked at in the light of the forgetful functor $$\Phi: (\text{2-groups)} \to (\text{groupoids})$$ which is a bifibration with a left adjoint say $D$, and a right adjoint $I$. As developed in Appendix B3 (Theorem B.3.2) of the book NAT, the cocartesian property of $\Phi$ is given by a pushout of 2-groupoids $$ \begin{matrix} D\Phi(K) &\xrightarrow{D(F)}& D(H) \\ \downarrow&& \downarrow \\ K & \to &F_*(K).\end{matrix} $$
The notion of ``free" 2-groupoid on $F: G \to H$ is the special case when $K=I(G)$. This is not stated in that Appendix but is the construction used in Part 1. For example if $F:P \to Q$ is a morphism of groups then the free crossed module on $F$ is the induced crossed module $F_*(P \to P)$, where the identity crossed module is $I(P)$. This generalises to free crossed modules over groupoids.