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(It seems you don’t mean to say “representations” but “central extensions”? There is a close relation, but the general topic of (“linear”) representations of higher groups is deep and murky, while the central extensions of higher groups are much better understood.)
Generally, extensions of an $\infty$-group $G$ by a braided $\infty$-group $A$ (both can be the coherent 2-groups that you are after, but may also be more general) are equivalently $\mathbf{B}A$-principal bundles over $\mathbf{B}G$.
(this is Section 4.3 in NSS12 here, or Section 3.6.14 in dcct here)
By the general classification of principal $\infty$-bundles, (Theorem 3.17 in NSS12 here) this means that these extensions are classified by maps
$\mathbf{B}G \longrightarrow \mathbf{B}^2 A$hence by
$\pi_0 \mathbf{H}(\mathbf{B}G, \mathbf{B}^2 A)$hence by
$H^2_{Grp}(G, A)$the “cohesive $\infty$-group degree-2 cohomology of $G$ with coeffcients in $A$”. Again, you may restrict this to coherent 2-groups if desired.
This holds for $G$ being group $\infty$-stacks, but to compute classifications you’ll want to restrict to controlled cases where, say, $G$ is a Lie group.
In the case that $G$ is a Lie group, the above reduces to Segal-Brylinski smooth group cohomology (Theorem 4.4.36 in dcct here),
Combining all this, one finds in particular the String 2-group extensions (Section 5.1.4 in dcct here )
The only systematic way, that I am aware of, to get the interesting 2-representations of 2-groups remains that in:
Appendix A “2-Vector spaces and the canonical 2-representation” of arXiv:0806.1079
As formulated there this invokes strict 2-group models of the given 2-group. This should be just a technical convenience, not a restriction. For the string 2-groups the strict 2-group model is established in arXiv:math/0504123.
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