  Let $L$ be an even positive definite lattice with no roots, i.e.,
$L(2)=\{x\in L\mid (x|x)=2\}=\emptyset$. 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 $V_L^{\hat{g}}$. As
our main result, we show that $Aut(V_L^{\hat{g}})$ is isomorphic to
$N_{Aut(V_L)}(\langle \hat{g}\rangle)/ \langle\hat{g}\rangle $ unless $L\cong
\sqrt{2}E_8$ or $BW_{16}$.
