  We study the propagation of massless fermionic fields in the background of a
three-dimensional Lifshitz black hole, which is a solution of conformal
gravity. The black hole solution is characterized by a null dynamical exponent.
Then, we compute analytically the quasinormal modes, the area spectrum, and the
absorption cross section for fermionic fields. The analysis of the quasinormal
modes shows that the fermionic perturbations are stable in this background. The
area and entropy spectrum are evenly spaced. At the low frequency limit, it is
observed that there is a range of values of the angular momentum of the mode
that contributes to the absorption cross section, whereas it vanishes at the
high frequency limit. In addition, by a suitable change of variables a
gravitational soliton can also be obtained and the stability of the quasinormal
modes are studied and ensured.
