$c$ is your crossed module, or 2-group, in a sense. Anything more concrete will depend on a choice of a cocycle description of the pointed set $[BH,B^3A]$.

For example, $[BH,B^3A]$ classifies central 2-group extensions $BA \to G \to H$ up to weak equivalence. Hence, one could regard such extensions as the cocycles, and then $c=G$.

A probably more concrete cocycle model is given by the smooth group cohomology $H^3_{sm}(H,A)$. By this I mean a smooth version of "Segal-Mitchison cohomology", like considered by Brylinksi. Recent references are:

• Schommer-Pries, Christopher J., Central extensions of smooth 2-groups and a finite-dimensional string 2-group, Geom. Topol. 15, No. 2, 609-676 (2011). ZBL1216.22005.

• Wagemann, Friedrich; Wockel, Christoph, A cocycle model for topological and Lie group cohomology, Trans. Am. Math. Soc. 367, No. 3, 1871-1909 (2015). ZBL1308.22006.

• Blanco, Jaider; Uribe, Bernardo; Waldorf, Konrad, Pontrjagin duality on multiplicative Gerbes, preprint.

In the first reference it is described how to obtain the cocycle from the 2-group extension. As usual, this will depend on choices of open covers, sections, etc., so that one cannot expect a canonical formula for $c$.

In several specific cases, $H^3_{sm}(H,A)$ can be identified with more familiar sets. For example, if $H$ is compact and $A=U(1)$, we have $H^3_{sm}(H,A)\cong H^4(BH,\mathbb{Z})$. If $G$ is the string 2-group, then $c \in H^4(BSpin,\mathbb{Z})=Z$ is a generator. If $G$ is the T-duality 2-group, then $c\in H^4(B\mathbb{T}^{2n},\mathbb{Z})$ is $c=\sum_{i=1}^n c_i \cup c_{i+n}$, where $c_i \in H^2(BS^1,\mathbb{Z})$ is the first Chern class of the $i$th factor. Some further examples have been computed in the above-mentioned preprint in Sections 5.4 - 5.6.

$c$ is your crossed module, or 2-group, in a sense. Anything more concrete will depend on a choice of a cocycle description of the pointed set $[BH,B^3A]$.

For example, $[BH,B^3A]$ classifies central 2-group extensions $BA \to G \to H$ up to weak equivalence. Hence, one could regard such extensions as the cocycles, and then $c=G$.

A probably more concrete cocycle model is given by the smooth group cohomology $H^3_{sm}(H,A)$. By this I mean a smooth version of "Segal-Mitchison cohomology", like considered by Brylinksi. Recent references are:

• Schommer-Pries, Christopher J., Central extensions of smooth 2-groups and a finite-dimensional string 2-group, Geom. Topol. 15, No. 2, 609-676 (2011). ZBL1216.22005.

• Wagemann, Friedrich; Wockel, Christoph, A cocycle model for topological and Lie group cohomology, Trans. Am. Math. Soc. 367, No. 3, 1871-1909 (2015). ZBL1308.22006.

• Blanco, Jaider; Uribe, Bernardo; Waldorf, Konrad, Pontrjagin duality on multiplicative Gerbes, preprint.

In the first reference it is described how to obtain the cocycle from the 2-group extension. As usual, this will depend on choices of open covers, sections, etc., so that one cannot expect a canonical formula for $c$.

In several specific cases, $H^3_{sm}(H,A)$ can be identified with more familiar sets. For example, if $H$ is compact and $A=U(1)$, we have $H^3_{sm}(H,A)\cong H^4(BH,\mathbb{Z})$. If $G$ is the string 2-group, then $c \in H^4(BSpin,\mathbb{Z})=\mathbb{Z}$ is a generator. If $G$ is the T-duality 2-group, then $c\in H^4(B\mathbb{T}^{2n},\mathbb{Z})$ is $c=\sum_{i=1}^n c_i \cup c_{i+n}$, where $c_i \in H^2(BS^1,\mathbb{Z})$ is the first Chern class of the $i$th factor. Some further examples have been computed in the above-mentioned preprint in Sections 5.4 - 5.6.