The book partially titled Nonabelian Algebraic Topology (NAT) has an extensive treatment of crossed modules and their role in cohomology and in homotopy classification.

A main difference from standard homological algebra texts is the use of crossed complexes and of free crossed resolutions of groups. In particular, some seemingly complicated cocycle conditions turn out to be related to a boundary rule in a standard free crossed complex; so the rule you give does, with the appropriate conventions, become $c\delta =0$.

The book partially titled Nonabelian Algebraic Topology (NAT) (pdf available) has an extensive treatment of crossed modules and their role in cohomology and in homotopy classification.

A main difference from standard homological algebra texts is the use of crossed complexes and of free crossed resolutions of groups. In particular, some seemingly complicated cocycle conditions turn out to be related to a boundary rule in a standard free crossed complex; so the rule you give does, with the appropriate conventions, become $c\delta =0$.

Background is given in the exposition `Groupoids and crossed objects in algebraic topology',Homology, homotopy and applications, 1 (1999) 1-78, available here. This contains some calculations of the $k$-invariants of some crossed modules.

Further calculations are available in the paper

G. Ellis and Le Van Luyen, "Homotopy 2-types of low order", available from the preprint list of Graham Ellis.