I am trying to understand the following from Principal Infinity Bundles - General by Nikolaus, Schreiber and Stevenson.

[![enter image description here][1]][1]

So following the reference there to Nikolaus-Waldorf tells us that given any (smoothly separable) Lie $2$-group (let's work in the model of crossed modules for convenience) $[d: H \to G]$, the following is a fiber sequence of Lie $2$-groups
$$[\ker d \to 1] \to [H \to G] \to [1 \to \text{coker }d]$$
What this means precisely is explained in more detail in Lemma 2.2 of this [paper][2]. In the notation of the problem, this corresponds to
$$BA \to G \to H$$
being a fiber sequence of $2$-groups, where $H$ is a group discretely embedded as a $2$-group and $A$ is an abelian group that is being delooped to a $2$-group. Now since $B$ is a right adjoint, it preserves this fiber sequence, and that gives us that
$$B^2 A \to BG \to BH$$ is a fiber sequence.

**So here are my questions**:
 - What is this map $c$? In Definition 4.30 of the same paper they give a definition of what a central extension of infinity groups should be involving this $c$, but I don't know how to construct it in this example.
 - In what ways can I present this map? Namely how would I present $B^3 A$? Can I use simplicial groups?
 - Why is this the internal and only nontrivial stage of the postnikov tower? 

  [1]: https://i.sstatic.net/eqMEr.png
  [2]: https://webusers.imj-prg.fr/~gregory.ginot/papers/CohLie2gp.pdf