The most general renormalizable Lagrangian of the Yukawa couplings among these fields and Majorana mass terms for the neutrinos that are invariant under both the standard model gauge group and the $A_{5}\times Z_{2}\times Z_{3}$ discrete flavor symmetry is $$\begin{aligned}
\label{A5Yukawa}
\emph{L}_{\rm {Yukawa}} &=& \frac{1}{2}M_{1}N^{(1)}_{R}N^{(1)}_{R} + \frac{1}{2}M_5N_{R5}N_{R5} + Y_{S1}(S_{4}N_{R5}N_{R5}) + Y_{S2}(S_{4}(l^{-}_{R3})^{c}l^{-}_{R5}) \nonumber \\
&+& Y_{1}(L_{L4}N^{(1)}_{R}H_{4}) + Y_{2}(L_{L4}N_{R5}H_{4}) + Y_{3}(L_{L4}N_{R5}\Phi_{3}) \nonumber \\ 
&+& Y_{4}(L_{L4}l_{R5}H'_{4}) + Y_{5}(L_{L4}l^{(1)}_{R1}H'_{4}) + Y_{6}(L_{L4}l^{(2)}_{R1} H'_{4}) + \rm {H.c.} \end{aligned}$$ Suppose the scalar $S_{4}$ develops a VEV $\langle S_{4} \rangle = (V_{S}, 0, 0, 0)$. It will break the discrete group $A_{5}$ into $A_{4}$, causing the irreps of $A_{5}$ to decompose as $$\begin{aligned}
&A_{5}& ~\rightarrow  A_{4}\nonumber \\
&\textbf{1}& \rightarrow \textbf{1} \nonumber \\
&\textbf{3}& \rightarrow \textbf{3} \nonumber \\
&\textbf{3}'& \rightarrow \textbf{3} \nonumber \\
&\textbf{4}& \rightarrow \textbf{1} + \textbf{3} \nonumber \\
&\textbf{5}& \rightarrow \textbf{1}' + \textbf{1}'' + \textbf{3} \nonumber\end{aligned}$$ Hence the lepton fields are decomposed as $$L_{L_{4}} \rightarrow L_{L1} (\textbf{1}, +1,\omega) + L_{L3} (\textbf{3},+1,\omega) \;\;,$$ $$l_{R5} \rightarrow l_{R3} (\textbf{3}, -1, +1) + l_{R1'} (\textbf{1}', -1, +1) + l_{R1''} (\textbf{1}'',-1, +1) \;\;,$$ and $$N_{R5} \rightarrow N_{R3} (\textbf{3},+1, +1) + N^{(2)}_{R} (\textbf{1}',+1, +1) + N^{(3)}_{R} (\textbf{1}'',+1, +1) \;\;.$$ Here we assume the right-handed $A_{4}$ triplet $l_{R3}$ is combined with the lepton $(l)^{c}_{R3}$ shown 