Pull back group cohomology onto handle decomposition

A construction encountered in the Dijkgraaf-Witten invariant uses the following ingredients:

• An oriented, (assumed here to be smooth) manifold $$M^n$$
• A finite group $$G$$ (and a field, chosen to be $$\mathbb{C}$$ here)
• An $$n$$-th cohomology class $$[\omega] \in H^n(G, U(1))$$ (where $$U(1)$$ is considered as a discrete group here)
• A homomorphism $$\phi\colon \pi_1(M) \to G$$

There is a canonical (up to homotopy) map $$c\colon M \to B\pi_1(M)$$. We can then abstractly construct the cohomology class $$c^* \phi^* ([\omega]) \in H^n(M, U(1))$$.

I'm interested describing this cohomology class very concretely. Assume I have given:

• $$M$$ as a handle decomposition (and consequently we consider Morse cohomology, where the $$k$$-th grade of the complex is generated by the $$k$$-handles)
• $$\phi$$ as an assignment of group elements to each 1-handle of $$M$$ (satisfying the relations for 2-handles)
• $$[\omega]$$ represented through a concrete cocycle $$\omega\colon G^n \to U(1)$$, i.e. a cocycle in the simplicial cohomology of $$BG$$

How can I explicitly pull back $$[\omega]$$ along $$\phi \circ c$$?

Some remarks:

• A handle decomposition of $$M$$ also gives a CW-complex, I think. So $$c\colon M \to B\pi_1(M)$$ can be chosen canonically as a cellular map via the Postnikov construction, and it's the pullback on cohomology can be derived from the pullback on the level of complexes.
• The trouble seems to be that $$\phi$$ isn't obviously cellular on higher cells if we describe $$BG$$ as a simplicial complex (which we might have to at some point because $$[\omega]$$ is given that way).