  In a series of papers \cite{KS,MR}, Krawitz, Milanov, Ruan, and Shen have
verified the so-called Landau-Ginzburg/Calabi-Yau (LG/CY) correspondence for
simple elliptic singularities $E_N^{(1,1)}$ ($N=6,7,8$). As a byproduct it was
also proved that the orbifold Gromov--Witten invariants of the orbifold
projective lines $\mathbb{P}^1_{3,3,3}$, $\mathbb{P}^1_{4,4,2}$, and
$\mathbb{P}^1_{6,3,2}$ are quasi-modular forms on an appropriate modular group.
While the modular group for $\mathbb{P}^1_{3,3,3}$ is $\Gamma(3)$, the modular
groups in the other two cases were left unknown. The goal of this paper is to
prove that the modular groups in the remaining two cases are respectively
$\Gamma(4)$ and $\Gamma(6)$.
