Fourvolutions and automorphism groups of orbifold lattice vertex operator algebras

Let $L$ be an even positive definite lattice with no roots, i.e., $L(2)=\{x\in L\mid (x\mathrm{\mid }x)=2\}=\mathrm{\varnothing }$. Let $g\in O(L)$ be an isometry of order $4$ such that $g^2=-1$ on $L$. In this article, we determine the full automorphism group of the orbifold vertex operator algebra $V_L^{\widehat{g}}$. As our main result, we show that $Aut(V_L^{\widehat{g}})$ is isomorphic to $N_{Aut(V_L)}(⟨\widehat{g}⟩)\mathrm{/}⟨\widehat{g}⟩$ unless $L\cong \sqrt{2}E_8$or$BW_{16}$.