  We prove two Lieb-Schultz-Mattis type theorems that apply to any
translationally-invariant and local fermionic $d$-dimensional lattice
Hamiltonians for which fermion-number conservation is broken down to the
conservation of fermion parity. We show that when the internal symmetry group
$G^{\,}_{f}$ is realized locally (in a repeat unit cell of the lattice) by a
nontrivial projective representation, then the ground state cannot be
simultaneously nondegenerate, symmetric (with respect to lattice translations
and $G^{\,}_{f}$), and gapped. We also show that when the repeat unit cell
hosts an odd number of Majorana degrees of freedom and the cardinality of the
lattice is even, then the ground state cannot be simultaneously nondegenerate,
gapped, and translationally symmetric.
