Can I recover a crossed module by its homomorphisms? This is a follow up to this question.

Imagine there is a finitely presented crossed module $\mathcal{G} = (G,H, -\triangleright-\colon G \to \operatorname{Aut}(H), \delta\colon H \to G)$ which I don't know. But for every other finite crossed module $\mathcal{G}'$, I know the number of homomorphisms $\mathcal{G} \to \mathcal{G}'$.
With these numbers, can I recover $\mathcal{G}$ up to isomorphism?

The motivation behind this is to understand the strength of the Yetter model. It counts homotopy equivalence classes from a 2-type to a fixed 2-type. Then a natural question is: "Given a manifold with an unknown homotopy 2-type, how much information about this 2-type can we recover from it through the Yetter model?"
By "finitely presented" or "finite" I mean that $G$ and $H$ are finitely presented, respectively, finite.
 A: I cannot be sure that there is no cunning method to do what you say but here is a comment about the problem.  
The numbers you mention only give the dimensions of various vector spaces of the Yetter model, they do not tell you how these vector spaces are related by the functoriality of the TQFT concerned. It seems to me that this could be viewed from the perspective of  Yoneda lemma type results. Suppose $\mathcal{C}$ is a category and $C$ an object, then you can get all the needed information on $C$ from the functor $\mathcal{C}(C,-): \mathcal{C}\to Sets$, but you cannot hope to do this by knowing just the sets $\mathcal{C}(C,D)$, you do need the functors concerned, or rather it would be interesting to characterise those categories $\mathcal{C}$ in which knowledge of those sets was sufficient.
You might argue that the fact of working with homotopy classes / natural transformations /  conjugacy of maps would tell you something about the functors but you are not even encoding the groupoid of maps under homotopy, merely the $\pi_0$ of that, if you see what I mean, so that is not going to get you very far.
