Given a short exact sequence $$ 1 \to B\mathbb{Z}_2 \to \mathbb{G} \to O(n) \to 1 $$ and the fiber sequence: $$ B^2\mathbb{Z}_2 \to B\mathbb{G} \to BO(n), $$ classified by $\beta \in H^3(BO(n), \mathbb{Z}_2)=(\mathbb{Z}_2)^3$.
Question: Would you call the $\mathbb{G}$ (there are $(\mathbb{Z}_2)^3$ choices), a group or 2-group, or both?
Why is $\mathbb{G}$ a group? What is the perspective?
Why is $\mathbb{G}$ a 2-group? What is the perspective?