Fermion masses and mixing from the double cover and metaplectic cover of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msub><mml:mi>A</mml:mi><mml:mn>5</mml:mn></mml:msub></mml:math> modular group

We perform a comprehensive study of the homogeneous finite modular group ${A}_{5}^{\ensuremath{'}}$ which is double covering ${A}_{5}$. The integral weight and level 5 forms have been constructed up to 6 they are decomposed into irreducible representations ${A}_{5}^{\ensuremath{'}}$. Then we systematical analysis models for lepton masses mixing. phenomenologically viable with minimal number free parameters results fit presented. find out 15 9 real can accommodate experimental data sector. After including generalized $CP$ symmetry, 7 found out. apply symmetry quark sector, quark-lepton unification model given. framework invariance extended include rational 5. ring at be generated by two algebraically independent $1/5$ denoted ${F}_{1}(\ensuremath{\tau})$ ${F}_{2}(\ensuremath{\tau})$. give expressions 3 arrange them multiplets metaplectic ${\stackrel{\texttildelow{}}{\mathrm{\ensuremath{\Gamma}}}}_{5}\ensuremath{\cong}{A}_{5}^{\ensuremath{'}}\ifmmode\times\else\texttimes\fi{}{Z}_{5}$. A neutrino mass ${\stackrel{\texttildelow{}}{\mathrm{\ensuremath{\Gamma}}}}_{5}$ presented, phenomenological predictions analyzed numerically.