  In this work, we study the inhomogeneous BMS free fermion theory, and show
that it gives a free field realization of the BMS Ising model. We find that
besides the BMS symmetry there exists an anisotropic scaling symmetry in BMS
free fermion theory. As a result, the symmetry of the theory gets enhanced to
an infinite dimensional symmetry generated by a new type of BMS-Kac-Moody
algebra, different from the one found in the BMS free scalar model. Besides the
different coupling of the $u(1)$ Kac-Moody current to the BMS algebra, the
Kac-Moody level is nonvanishing now such that the corresponding modules are
further enlarged to BMS-Kac-Moody staggered modules. We show that there exists
an underlying $W(2,2,1)$ structure in the operator product expansion of the
currents, and the BMS-Kac-Moody staggered modules can be viewed as
highest-weight modules of this $W$-algebra. Moreover we obtain the BMS Ising
model by a fermion-boson duality. This BMS Ising model is not a minimal model
with respect to BMS$_3$, since the minimal model construction based on BMS Kac
determinant always leads to chiral Virasoro minimal models. Instead, the
underlying algebra of the BMS Ising model is the $W(2,2,1)$-algebra, which can
be understood as a quantum conformal BMS$_3$ algebra.
