If 1 G I> E X-4 I -> 1 is a group extension, with t an inclusion, any automorphism T of E which takes G onto itself induces automorphisms T on G and a on 11. However, for a pair (a, T) of automorphism of 11 and G, there may not be an automorphism of E inducing the pair. Let Xx: H -IOut G be the homomorphism induced by the given extension. A pair (a, T) E Aut H1 x Aut G is called compatible if a fixes ker a, and the automorphism induced by a on WIIo is the same as that induced by the inner automorphism of Out G determined by T-. Let C C -H2(H, ZG). The last map is not surjective in general. It is not even a group homomorphism, but the sequence is nevertheless "exact" at C in the obvious sense. 1. Notation. If G is a group with subgroup H, we write H< G; if H is normal in G, H