respectively. Take two groups $G$ and $G'$ so that $J_{F_A} \subseteq G \subseteq SL(F_A)$ and $J_{F_{A^T}} \subseteq G^T \subseteq SL(F_{A^T})$. We then obtain two Calabi-Yau orbifolds $Z_{A,G} : = X_A/\tilde G$ and $Z_{A', G'} : = X_{A'} / \tilde G '$.

Even if these two orbifolds are in the same family of Calabi-Yau varieties, they may have BHK mirrors that are not in the same quotient of weighted-projective $n$-space (see Section 5 for an explicit example or Tables 5.1-3 of [@DG]). Take the polynomials $$F_{A^T} = \sum_{i=0}^n \prod_{j=0}^n x_j^{a_{ji}}; \qquad F_{(A')^T} = \sum_{i=0}^n \prod_{j=0}^n x_j^{a_{ji}'}.$$ They are quasihomogeneous polynomials but not necessarily with the same weights. So they cut out hypersurfaces $X_{A^T}$ and $X_{(A')^T}$. Take the dual groups $G^T$ and $(G')^T$ to $G$ and $G'$, respectively. We quotient each hypersurface by their respective groups, $\tilde G^T := G^T/ J_{F_{A^T}}$ and $(\tilde G')^T : = (G')^T / J_{F_{(A')^T}}$. We then have the BHK mirror dualities: $$\begin{aligned}
Z_{A,G} &\stackrel{\text{BHK mirrors}}{\longleftrightarrow} Z_{A^T, G^T} \\
Z_{A',G'} &\stackrel{\text{BHK mirrors}}{\longleftrightarrow} Z_{(A')^T, (G')^T} 
\end{aligned}$$

In this section, we will investigate and compare the birational geometry of the BHK mirrors of the Calabi-Yau orbifolds $Z_{A, G}$ and $Z_{A',G'}$ by using the Shioda maps. Take positive integers $d$ and 