On k-invariants of crossed modules corresponding to the dihedral groups

Any group G gives rise to a crossed module ∂ : G→ Aut(G), where ∂(g) is the inner automorphism defined by x↦ gxg⁻¹. This crossed module is denoted by $AUT(G)$. The first k-invariant of the classifying space of $AUT(G)$ provides the class kAUT(G)∈ H³(Out(G), Z(G)), where Z(G) is the centre of G and $Out(G)$ is the group of outer automorphisms of G. The goal of this work is to investigate whether this class is trivial for dihedral groups and their generalizations. Specifically, we compute the class kAUT(Dₙ) explicitly when n is power of two.