  We propose an SU(5) GUT model extended with two additional pairs of
$\mathbf{10}$ representation vector-like fermions. The CDF collaboration $W$
boson mass anomaly is explained by using the VEV of a real $\mathrm{SU(2)_L}$
triplet scalar coming from the $\mathbf{24}$ representation Higgs. The
vector-like fermions are decomposed partly into vector-like quark doublets.
Those vector-like quark doublets acquire mass from two sources; through the
Yukawa interaction with the real $\mathrm{SU(2)_L}$ triplet via a
type-$\rm{I\hspace{-.01em}I}$ seesaw-like mechanism. And, they acquire mass
from the $\mathbf{24}$ representation Higgs. We assume that the mass for the
vector-like quark doublets is expressed in terms of the real triplets mass. By
combining the constraints on the vector-like quark masses with those on the
heavy Higgs boson masses, we can obtain the narrow allowed mass ranges for the
vector-like quark doublet and the real triplet. Therefore, our model can be
tested in searches for these particles in the near future. In addition, the two
additional pairs of vector-like fermions allow the SM gauge couplings to unify
successfully at $M_{\mathrm{GUT}}\thickapprox5.1\times10^{15}$~GeV. Our model
is also testable by the future Hyper-Kamiokande experiment via the proton decay
lifetime $\tau_p(p\to\pi^0{e^+})<1.0\times10^{35}$~years.
