equations of different pencils of quintics in ${\mathbb{P}}^4$. The Shioda maps were then further generalized to look at GKZ hypergeometric systems for certain families of Calabi-Yau varieties in weighted-projective space in [@Bi]. This paper provides a more concrete description of how Shioda maps relate to BHK mirror symmetry than the previous two papers, and explains the groups used in the theorems of [@BvGK] and [@Bi] in the context of BHK mirrors (see Section 3.2). In future work, this framework will be used to probe Kähler moduli space of the Calabi-Yau orbifolds.

[**Organization of the Paper.**]{} In Section 2, we review the BHK mirror construction and the results of Chiodo and Ruan. In Section 3, we use the Shioda map to discuss the birational geometry of the BHK mirrors and the groups involved in [@BvGK] and [@Bi]. We then show the birationality of the Calabi-Yau orbifolds $Z_{A,G}$ and $Z_{A^T, G^T}$ to finite quotients of a Fermat variety in projective space. In Section 4, we use the results found in Section 3 to prove Theorem \[main\]. Section 5 concludes the paper by giving an explicit example where we take two Calabi-Yau orbifolds and show that their BHK mirrors are hypersurfaces of different 