  In this article, we study the automorphism group of the cyclic orbifold of a
vertex operator algebra associated with a rootless even lattice for a lift of a
fixed-point free isometry of odd prime order $p$. We prove that such a cyclic
orbifold contains extra automorphisms, not induced from automorphisms of the
lattice vertex operator algebra, if and only if the rootless even lattice can
be constructed by Construction B from a code over $\mathbb{Z}_p$ or is
isometric to the coinvariant lattice of the Leech lattice associated with a
certain isometry of order $p$.
