the other three SM neutrinos, hence they can be used to easily satisfy the experimental bound $m_{\nu_4} > M_{Z}/2$. At leading order the three light neutrino masses are $m_{\nu_1} = m_{\nu_3}$ and $m_{\nu_2} = 0$, which are phenomenological unacceptable. However, the inclusion of perturbative effects on the masses due to the heavy Majorana neutrinos through the VEVs of $S_{3}$ will correct the above three light neutrinos masses and allow a good fit to the oscillation data. We note that this perturbative effect is from the assumed $A_5$ symmetry and its subsequent spontaneous break down to $A_4$. The situation is similar to the proposal in Ref. [@Ma:2004zv], in which the neutrino masses can be split by the small VEVs of several heavy Higgs triplets which are assigned to different representations of the $A_{4}$ group. Therefore, taking the VEVs of $S_{3}$, to be $\langle S_{3} \rangle = (\delta_1,\delta_2,\delta_3)$, the perturbations to the right-handed Majorana mass matrix, as shown in Eq. (\[MR\]) are $$\begin{aligned}
\label{MRp}
M_{N_{R}} = \left(\begin{array}{cccccc}M_{1} & 0 & 0 & 0 & 0 & 0 \\0 & 0 & M_{2} & 0 & 0 & 0 \\0 & M_{2} & 0 & 0 & 0 & 0 \\0 & 0 