Integral Cohomology and Detection of $w$-basic 2-groups

In this paper we compute the cohomology $H^*(G;Z^w)$ of all $w$-basic 2-groups $(G,w)$ with integral coefficients twisted by the orientation character $w$. We also calculate appropriate restiction maps and thus prove that the cohomology of any $w$-basic group is detected by subgroups isomorphic to one of five types, and we provide a sample application of this main theorem.