I would like to have a reference about the construction and properties of $H^1(X;G)$ for $X$ a topological space and $G$ a non-abelian group (in the spirit of expanding and clarifying the first rows of the first comment to this).
Connections of that with the theory developed in Hatcher's AT Section 4.3, i.e. correspondences between $H^1(X;G)$ and maps $\langle X,K(G,1)\rangle$ (which in his treatment involves only abelian groups) are also very welcome.
Thank you in advance.
The book Nonabelian Algebraic Topology deals even with coefficients in a crossed module and uses techniques in the book to calculate some homotopy classes of maps: see Section 12.3.
The background is that to get some nonabelian coefficients it is desirable to have some nonabelian type of chains, and a first candidate seems to be crossed complexes.
Let me know of any further questions on this.