  Let $G$ be a non-amenable countable group. We show that every almost
automorphic $G$-action on a compact Hausdorff space, with a maximal
equicontinuous factor whose phase space is a Cantor set, admits invariant
probability measures (this partially answers a question posed by Veech). In
particular, every Toeplitz $G$-subshift has a non-empty space of invariant
measures, meaning that this family of subshifts is not a test for amenability
for countable groups. We prove that almost one-to-one extensions without
measures ensure the existence of symbolic almost one-to-one extensions with
equal characteristics. As a consequence, we obtain the most general result of
this paper. Finally, as a corollary of our results, we deduce that the class of
Toeplitz subshifts is not dense in the space of infinite transitive subshifts
of $\Sigma^G$, unlike $G=\mathbb{Z}$.
