Cell structure on $B\mathbb{G}$ and the bar resolution of $\mathbb{G}$

Consider $\mathbb{G}$, which can be viewed as a group, as well as a 2-group.

(For example, given a short exact sequence $$1 \to BG_2 \to \mathbb{G} \to G_1 \to 1$$ and the fiber sequence: $$B^2G_2 \to B\mathbb{G} \to BG_1.$$ where $\mathbb{G}$ can be viewed as a group or a 2-group.)

The consideration is the following:

$$G \text{ has one object}.$$

$$BG \text{ has one 0-cell}.$$

$$1\text{-morphisms} = \text{ elements of } \mathbb{G}.$$

$$1\text{-cells} = 1\text{-morphisms of } \mathbb{G}.$$

• Question: What is the precise relation between the cell structure on $B\mathbb{G}$ and the bar resolution of $\mathbb{G}$?