  In this paper, we study the quasinormal modes of the massless Dirac field for
charged black holes in Rastall gravity. The spherically symmetric black hole
solutions in question are characterized by the presence of a power-Maxwell
field, surrounded by the quintessence fluid. The calculations are carried out
by employing the WKB approximations up to the thirteenth order, as well as the
matrix method. The temporal evolution of the quasinormal modes is investigated
by using the finite difference method. Through numerical simulations, the
properties of the quasinormal frequencies are analyzed, including those for the
extremal black holes. Among others, we explore the case of a second type of
extremal black holes regarding the Nariai solution, where the cosmical and
event horizon coincide. The results obtained by the WKB approaches are found to
be mostly consistent with those by the matrix method. It is demonstrated that
the black hole solutions for Rastall gravity in asymptotically flat spacetimes
are equivalent to those in Einstein gravity, featured by different asymptotical
spacetime properties. As one of its possible consequences, we also investigate
the behavior of the late-time tails of quasinormal models in the present model.
It is found that the asymptotical behavior of the late-time tails of
quasinormal modes in Rastall theory is governed by the asymptotical properties
of the spacetimes of their counterparts in Einstein gravity.
