  Experimental data on the neutrino mixing and masses strongly suggest an
underlying approximate symmetry of the relevant Yukawa superpotential terms.
Intensive phenomenological explorations during the last decade indicate that
permutation symmetries such as S_4, A_4 and their subgroups, under certain
assumptions and vacuum alignments, predict neutrino mass textures compatible
with such data. Motivated by these findings, in the present work we analyse the
neutrino properties in F-theory GUT models derived in the framework of the
maximal underlying E_8 symmetry in the elliptic fibration. More specifically,
we consider local F-SU(5) GUT models and study in detail spectral cover
geometries with monodromies associated to the finite symmetries S_4, A_4 and
their transitive subgroups, including the dihedral group D_4 and Z_2 X Z_2. We
discuss various issues that emerge in the implementation of S_4, A_4 neutrino
models in the F-theory context and suggest how these can be resolved. Realistic
models are presented for the case of monodromies based on their transitive
subgroups. We exemplify this procedure with a detailed analysis performed for
the case of Z_2 X Z_2 model.
