Lifting of outer actions of groups

Abstract

Let G and N be arbitrary groups. In this paper, we construct an associated group extension ℰ_χ of N/Z(N) by G for any group homomorphism χ: G → OutN, and prove that χ can be lifted to a group action, that is, a group homomorphism from G to AutN, if and only if the extension ℰ_χ splits. Furthermore we obtain an explicit description for all such lifting homomorphisms and the number of its conjugacy classes, and give an application of the lifting technique of outer actions to the theory of group extensions.