  We predict the low energy signatures of a Flipped $SU(5) \times U(1)_{\chi}$
effective local model , constructed within the framework of F$-$theory based on
$ A_{4} $ symmetry. The Flipped SU(5) model from F Theory in the field of
particle physics is prominent due to its ability to construct realistic
four$-$dimensional theories from higher$-$dimensional compactifications which
necessitates a unified description of the fundamental forces and particles of
nature, used for exploring various extensions of the Standard Model. We study
Flipped $SU(5) \times U(1)_{\chi}$ Grand Unified Theories (GUTs) with $ A_{4} $
modular symmetry. In our model due to different modular weights assignments,
the fermion mass hierarchy exists with different weighton fields. The
constraints on the Dirac neutrino Yukawa matrix allows a good tuning to quark
and charged lepton masses and mixings for each weighton field, with the
neutrino masses and lepton mixing well determined by the type I seesaw
mechanism which occurs at the expense of some tuning which manifests itself in
charged lepton flavour violating decays which we explore here. The minimal
Flipped $SU(5$) model is supplemented with an extra right$-$handed type and its
complex conjugate electron state, $ E_{c} + \bar{E_{c}} $, as well as neutral
singlet fields. The $ E_{c} + \bar{E_{c}} $ pair gets masses of the order of
TeV which solves the $ g_{\mu}- 2$ discrepancy. The predictions of the model
for charged lepton flavour violation decay rate and proton decay could be
tested in near future experiments. Also we detect in our model the existence of
neutralino, its charge mass and spin via direct and indirect detection.
