Second nonabelian group cohomology: cocycles vs. gerbes

The cocycle data which you review together is a map of 2-groupoids $B \Gamma \to B Aut(B G)$ to the delooping of the automorphism 2-group "of $G$" (really: of $BG$). As for any cocycle with coefficients in an automorphism group, there is the corresponding associatived 2-bundle, hence a $BG$-fiber bundle. That's the corresponding Giraud G-gerbe and that's essentially the group extension.

There are some details on this spelled out in the nLab entry nonabelian group cohomology, though maybe that needs another touch.

The general abstract story of nonabelian cocycle, infinity-gerbes and associated infinity-bundles is in section 4 of

• Thomas Nikolaus, Urs Schreiber, Danny Stevenson, Principal infinity-bundles - General theory (arXiv:1207.0248, web)

I am not sure if this is the most general definition, but my proposal for 2nd nonabelian cohomology is combinatorial descent data in a cosimplicial crossed gropoid, modulo gauge equivalence. This is a development of an earlier idea of L. Breen.

This is worked out in my paper

Combinatorial descent data for gerbes, Journal of Noncommutative Geometry Volume 8, Issue 4, 2014, pp. 1083–1099 arXiv:1109.1919

For a geometric application see the paper

Twisted Deformation Quantization of Algebraic Varieties, Advances in Mathamatics, Volume 268, 2 January 2015, Pages 241–305 arXiv:0905.0488

Nonabelian $H^2$ in Galois cohomology can be defined in terms of: (1) cocycles, (2) extensions, (3) gerbes. The relations between these three definitions are described in Section 2.2 of Le principe de Hasse pour les espaces homogènes : réduction au cas des stabilisateurs finis by Cyril Demarche and Giancarlo Lucchini Arteche.