Could someone please give a reference about how to use Fox Calculus to compute the cohomology of a $2$-group $G$ with coefficients in a submodule of $\oplus^n F_2[G]$.
Is there a formula to count Fox derivations? $G\to F_2[G]$.
A 2-group? You are tough cookie. For a knot group, Crowell-Fox's book on knot theory does a nice job. In general its just an observation about the module structure of the CW groups of a normal covering space of your CW complex, where you are working over the group ring of the deck transformations. It only works for computing the 0th and 1st homology groups. Maybe you meant two complex. :) In which point it works for every homology group.