I think what you are looking for at least in terms of mappings rather than sections is Theorem 12.2.6 of the book Nonabelian AlgebraicTopology (NAT), which is phrased in terms of the homotopy theory of crossed complexes. Maybe this needs to be developed in terms of sections.
(Obstruction Class Theorem) Let $n \geqslant 2$ and let $F,C$ be reduced crossed complexes such that $F$ is free, $C$ is $n$-aspherical, and $C_i=0$ for $i >n$. Let $G= \pi_1 F, H= \pi_1 C, M=$ Ker $ \delta_n \colon C_n \to C_{n-1}$. Let $\theta \colon G \to H$ be a morphism of groups. Then there is defined an element $k_\theta \in H^{n+1}_{\theta\phi}(F,M)$, called the obstruction class of $\theta$, such that the vanishing of $k_\theta$ is necessary and sufficient for $\theta$ to be realised by a morphism $F \to C$.
If $k_\theta=0$, then the set $[F,C;\theta\phi]$ of homotopy classes of morphisms $F\to C$ realising $\theta \phi$ is bijective with $H^n_{\theta\phi}(F,M)$. $\square$
A reduced crossed complex $C$ is one which $C_0$ is a singleton.
The NAT book also spells out the adjoint relation of crossed complexes and chain complexes with a groupoid of operators. All this is related to the answer of S. carmeli.
The other author you could look at is Johannes Huebschmann, who developed work on crossed complexes and group cohomology, which also has relations to earlier work of A.S.-T. Lue on cohomomology with respect to a variety, see references [Hue...], and [Lue81], in NAT.