  We consider the general $\mathbb{Z}_2$-symmetric free-fermion model on the
finite periodic lattice, which includes as special cases the Ising model on the
square and triangular lattices and $\mathbb{Z}_n$-symmetric BBS
$\tau^{(2)}$-model with $n=2$. Translating Kaufman's fermionic approach to
diagonalization of Ising-like transfer matrices into the language of Grassmann
integrals, we determine the transfer matrix eigenvectors and observe that they
coincide with the eigenvectors of a square lattice Ising transfer matrix. This
allows to find exact finite-lattice form factors of spin operators for the
statistical model and the associated finite-length quantum chains, of which the
most general is equivalent to the XY chain in a transverse field.
