  Although quasi-dilaton massive gravity is a well-defined gravitational
theory, it exhibits instabilities and suffers from the strong coupling problem.
In this work we construct an extension of the theory, through the inclusion of
the aether field. Focusing on flat Friedmann-Lemaitre-Robertson-Walker
geometry, we show the existence of exact, self-accelerating solutions at the
background level, characterized by an effective cosmological constant arising
from the graviton mass. Additionally, we perform a detailed perturbation
analysis, investigating separately the tensor, vector, and scalar
perturbations, extracting the dispersion relation of gravitational waves, and
determining the stability conditions for vector and scalar sectors. As we show,
there are always regions in the parameter space in which the obtained solutions
are free from ghost instabilities, as well as from the strong coupling problem.
Hence, although the aether field does not play an important role in the
background self-accelerating solutions, it does play a crucial role in the
alleviation of the perturbation-related problems of the simple quasi-dilaton
massive gravity.
