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.

otherfinite crossed module. Have you tried thinking of this for groups rather than crossed modules? That might give some insight. $\endgroup$ – Tim Porter Jun 30 '17 at 14:35