  We consider realistic F-theory GUT models based on discrete family symmetries
$A_4$ and $S_3$, combined with $SU(5)$ GUT, comparing our results to existing
field theory models based on these groups. We provide an explicit calculation
to support the emergence of the family symmetry from the discrete monodromies
arising in F-theory. We work within the spectral cover picture where in the
present context the discrete symmetries are associated to monodromies among the
roots of a five degree polynomial and hence constitute a subgroup of the $S_5$
permutation symmetry. We focus on the cases of $A_4$ and $S_3$ subgroups,
motivated by successful phenomenological models interpreting the fermion mass
hierarchy and in particular the neutrino data. More precisely, we study the
implications on the effective field theories by analysing the relevant
discriminants and the topological properties of the polynomial coefficients,
while we propose a discrete version of the doublet-triplet splitting mechanism.
