with $i=1,2,3,4$ represents the left-handed lepton doublets of the $i^{\text{th}}$ family and $\omega = e^{i2\pi/3}$ is the cubic root of unity. For the right-handed sector we introduce the charged lepton fields $$\begin{aligned}
\label{leprt}
l_{R5} (\textbf{5}, -1, +1) \quad, \quad  l^{c}_{R3} (\textbf{3}, -1, +1) \quad {\rm and} \quad l^{(1)}_{R1} (\textbf{1}, -1, +1) \quad, \quad l^{(2)}_{R1} (\textbf{1}, -1, +1) \end{aligned}$$ which transform as a $\bf 5$, a $\bf 3$, and two trivial singlets $\bf 1$ under $A_{5}$ symmetry respectively. Here $c$ stands for charge conjugation. The right-handed neutrinos are assigned as one quintet $\textbf{5}$ and a trivial singlet $$\begin{aligned}
\label{nu}
N_{R5} (\textbf{5}, +1, +1) \quad {\rm and} \quad N^{(1)}_{R} (\textbf{1}, +1, +1) \;\;. \end{aligned}$$ We have implicitly embedded the right-handed charged leptons $\mu_{R}$, $e_{R}$ in the 5 dimensional irreducible representation (irrep) of $A_{5}$.

For the Higgs sector we introduce three $A_{5}$ quartet and one $A_{5}$ triplet scalars. The first quartet $S_{4}$ is a gauge singlet which transform as $(\textbf{4},+1,+1)$ under $A_{5}\times Z_{2}\times Z_{3}$. The other two quartets are electroweak doublets with $Z_2 \times Z_3$ charges and are written as $H_{4} (\textbf{4}, +1,\omega^2)$ and $H'_{4} (\textbf{4}, -1,\omega^2)$. The $A_5$ triplet $\Phi_{3}(\textbf{3}, +1, \omega^2)$ is also an electroweak doublet and has nontrivial $Z_3$ charge. 