  The automorphism group of the vertex operator algebra $V_L^+$ is studied by
using its action on isomorphism classes of irreducible $V_L^+$-modules. In
particular, the shape of the automorphism group of $V_L^+$ is determined when
$L$ is isomorphic to an even unimodular lattice without roots, $\sqrt2R$ for an
irreducible root lattice $R$ of type $ADE$ and the Barnes-Wall lattice of rank
16.
