  We study the quasinormal modes of fermionic perturbations for an
asymptotically Lifshitz black hole in 4-dimensions with dynamical exponent z=2
and plane topology for the transverse section, and we find analytically and
numerically the quasinormal modes for massless fermionic fields by using the
improved asymptotic iteration method and the Horowitz-Hubeny method. The
quasinormal frequencies are purely imaginary and negative, which guarantees the
stability of these black holes under massless fermionic field perturbations.
Remarkably, both numerical methods yield consistent results; i.e., both methods
converge to the exact quasinormal frequencies; however, the improved asymptotic
iteration method converges in a fewer number of iterations. Also, we find
analytically the quasinormal modes for massive fermionic fields for the mode
with lowest angular momentum. In this case, the quasinormal frequencies are
purely imaginary and negative, which guarantees the stability of these black
holes under fermionic field perturbations. Moreover, we show that the lowest
quasinormal frequencies have real and imaginary parts for the mode with higher
angular momentum by using the improved asymptotic iteration method.
