The molecular-based magnetic materials AFeIIFeIII (C2O4)3 have a honeycomb structure in which FeII (S = 2) and FeIII (S = 5/2) occupy sites alternately. They can be described as mixed spin-2 and spin-5/2 Ising model with ferrimagnetic interlayer coupling. The influences of the transverse field on the internal energy and the specific heat of the molecular-based magnetic system have been studied numerically by using the effective-field theory with self-spin correlations and the differential operator technique.