Integral Cohomology and Detection of w-basic 2-groups 
Kimberly Pearson
Department of Mathematics, Indiana University, Bloomington, Indiana 47405 
Math. Comp. 65 (1996), pp. 291-306 

Abstract. 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.