Lifts of automorphisms of vertex operator algebras in simple current extensions

In this article, we study isomorphisms between simple current extensions of a simple VOA. For example, we classify the isomorphism classes of simple current extensions of the VOAs V-root 2E8(+) and V-Lambda 16(+), where Lambda(16) is the Barnes-Wall lattice of rank 16. Moreover, we consider the same simple current extension and describe the normalizer of the abelian automorphism group associated with this extension. In particular, we regard the moonshine module V(sic) as simple current extensions of five subVOAs V-L(+) for 2-elementary totally even lattices L, and describe corresponding five normalizers of elementary abelian 2-group in the automorphism group of V in terms of V-L(+). By using this description, we show that three of them form a Monster amalgam.