Automorphism groups of cyclic orbifold vertex operator algebras associated with the Leech lattice and some non-prime isometries

We determine the automorphism groups of the cyclic orbifold vertex operator algebras associated with coinvariant lattices of isometries of the Leech lattice in the conjugacy classes $4C,6E,6G,8E$ and $10F$. As a consequence, we have determined the automorphism groups of all the $10$ vertex operator algebras in [H"o], which are useful to analyze holomorphic vertex operator algebras of central charge $24$.