BHK mirror symmetry for K3 surfaces with non-symplectic automorphism

In this paper we consider the class of K3 surfaces defined as hypersurfaces in weighted projective space, and admitting a non-symplectic automorphism of non-prime order, excluding the orders 4, 8, and 12. We show that on these surfaces the Berglund-H"ubsch-Krawitz mirror construction and mirror symmetry for lattice polarized K3 surfaces constructed by Dolgachev agree; that is, both versions of mirror symmetry define the same mirror K3 surface.