  Motivated by experimental measurements indicating deviations from the
Standard Model predictions we discuss F-theory inspired models, which, in
addition to the three chiral generations contain a vector-like complete fermion
family. The analysis takes place in the context of $SU(5)\times U(1)'$ GUT
embedded in an $E_8$ covering group which is associated with the (highest)
geometric singularity of the elliptic fibration. In this context, the $U(1)'$
is a linear combination of four abelian factors subjected to appropriate
anomaly cancellation conditions. Furthermore, we require universal $U(1)'$
charges for the three chiral families and different ones for the corresponding
fields of the vector-like representations. Under the aforementioned
assumptions, we find 192 such models which can be classified into five distinct
categories with respect to their specific GUT properties. We exhibit
representative examples for each such class and construct the superpotential
couplings and the fermion mass matrices. We explore the implications of the
vector-like states in low energy phenomenology including the predictions
regarding the B-meson anomalies. The r\^ole of R-parity violating terms
appearing in some particular models of the above construction is also
discussed.
