Extensions of tensor products of ${\mathbb{Z}}_p$-orbifold models of the lattice vertex operator algebra $V_{\sqrt{2}{A}_{p-1}}$

Let $p$ be an odd prime and let $\widehat{\sigma }$ be an order $p$ automorphism of $V_{\sqrt{2}{A}_{p-1}}$ which is a lift of a $p$-cycle in the Weyl group $\mathrm{W}\mathrm{e}\mathrm{y}\mathrm{l}(A_{p-1})\cong {\mathfrak{S}}_p$. We study a certain extension $V$ of a tensor product of finitely many copies of the orbifold model $V_{\sqrt{2}{A}_{p-1}}^{⟨\widehat{\sigma }⟩}$ and give a criterion for $V$ that every irreducible $V$-module is a simple current.