As a natural extension of my previous question Higher Dijkgraaf-Witten Theory on DW Theory for a 1-form symmetry, we can extend now to 2-groups.
How can we generalize the notion of gauging to a 2-group? What does homotopy theory tells me about classifying maps $\gamma: M \rightarrow B\mathbb{G}$ where $B\mathbb{G}$ is a fibration $K(H,2)\rightarrow B\mathbb{G} \rightarrow K(G,1)$? Is there an explicit method to count $[X,B\mathbb{G}]$?
Assuming a trivial action between the two groups of the 2-group can we get a nice treatment? Clearly we already know that classifying maps to the fiber and to the base space are classified by the second and first cohomology groups $H^2(M,H)$ and $H^1(M,G)$ for $\mathbb{G}=(G,H,\alpha,\beta)$ and $\beta \in H^3(BG,H)$.
