Finite groups with small automizers of their nonabelian subgroups

Abstract

Let G be a group and let H be a subgroup of G. The automizer AutG(H) of H in G is defined as the group of automorphisms of H induced by conjugation of elements of NG(H). Thus AutG(H)≅NG(H)/CG(H), and we obviously have $$\tfrm{In(}H\tfrm{)≤Aut}_{G}\tfrm{(}H\tfrm{)≤Aut(}H\tfrm{).}$$ We call AutG(H) large if AutG(H)=Aut(H) and small if AutG(H)=In(H).