The (outer) automorphism group of a group extension

If K --> G --> Q is a group extension, then any automorphism of G which sends K into itself, induces automorphisms respectively on K and on Q. This subgroup of automorphisms of G is denoted by Ant (G, K) and is called the automorphism group of the extension K --> G --> Q. After establishing an interesting group action of Ant (K) x Ant (Q) on the set H-2(Q, K) of all 2-cohomology classes of Q with coefficients in K, a full description of Aut(G, K) and Out(G,K) = Aut(G,K)/(Inn(G)) is obtained in terms of various commutative diagrams. This picture is as general as possible, hence covering and further complementing similar ideas developed earlier by C. Wells ([5]), P. Conner & F. Raymond ([1]), D.J.S. Robinson ([3], [4]) and the author ([2]).