On comparing the cohomology of algebraic groups, finite Chevalley groups and Frobenius kernels

Let G be a semisimple simply connected algebraic group defined and split over the field image with p elements, image be the finite Chevalley group consisting of the image-rational points of G where q=pr, and Gr be the rth Frobenius kernel of G. This paper investigates relationships between the extension theories of G, image, and Gr over the algebraic closure of image. First, some qualitative results relating extensions over image and Gr are presented. Then certain extensions over image and Gr are explicitly identified in terms of extensions over G.