Central automorphisms of semidirect products

In this paper we describe the structure of AutZ/N(G) for a group G = HK, where K is a normal subgroup of G and N = H?K is AutZ(G)-invariant, in particular, if N = 1, this amounts to a description of the central automorphism group of the semi-direct product G = K x H. We also show that if N ? G and ?K(H/N) = N, then AutZ N(G) is a split extension. Particular if G is solvable, then AutZ/N(G) is an abelian by abelian split extension. This description of the group of central automorphisms of semidirect products is of great importance, because any solvable group has a splitting quotient.