A few questions about the equivalence between 2-types and crossed modules. For simplicity, assume everything is connected.

What is the precise statement? Is there an equivalence of categories (or at least a bijection of isomorphism classes) of the form

$$\{\text{2-truncated spaces}\}[\text{weak homotopy equivalence}^{-1}] \simeq \{\text{crossed modules}\}[\mathcal{W}^{-1}]$$

for some class of morphisms $\mathcal W$? If so, what is $\mathcal W$? Is it exactly the isomorphisms? What is a precise reference?

Crossed modules $1 \to A \to H_2 \to H_1 \to G \to 1$ are classified by $H^3(G;A)$ up to zigzags of morphisms of extensions. Does this translate to a classification of connected homotopy 2-types $X$ with $\pi_1(X) = G$ and $\pi_2(X) = A$ up to homotopy equivalence, or is it only up to a coarser equivalence relation? If this is a classification up to homotopy equivalence, then how does one see that a non-invertible morphism of crossed modules induces a homotopy equivalence of classifying spaces?

In any event, from every crossed module $1 \to A \to H_2 \to H_1 \to G \to 1$, I can extract an element of $H^3(G;A)$. Homotopically, this corresponds to a Postnikov invariant for a possibly non-principal Postnikov tower. Where is the theory of non-principal Postnikov invariants written, and in particular does this invariant (exist and) completely classify a 2-type?