Automorphisms and cohomology

Abstract: Let 1-> H -> G > Q -> 1 be an exact sequence of groups. In the paper of R. Oliver and J. Ventura, TAMS,362(2009), the following exact sequence was developed for centric extensions, i.e the centralizer of H in G is contained in H, 0-> H^1(Q,zH) -> Aut(G,H) -> N{Out H}(F Q)/F Q -> H^2(Q,zH) where Aut(G,H) are the automorphisms of G which restrict to an automorphism of H, F:Q -> Out H is the outer action determined by the extension, zH is the center of H with Q-action coming from F and N_{Out H} the normalizer. It is the aim of this paper to generalize the above sequence to arbitrary extensions, show how the above result is derived from the general exact sequence and derive other consequences of the general result including determining solvability of Aut(G,H).