  We show that every intermediate $\beta$-transformation is topologically
conjugate to a greedy $\beta$-transformation with a hole at zero, and provide a
counterexample illustrating that the correspondence is not one-to-one. This
characterisation is employed to (1) build a Krieger embedding theorem for
intermediate $\beta$-transformation, complementing the result of Li, Sahlsten,
Samuel and Steiner [2019], and (2) obtain new metric and topological results on
survivor sets of intermediate $\beta$-transformations with a hole at zero,
extending the work of Kalle, Kong, Langeveld and Li [2020]. Further, we derive
a method to calculate the Hausdorff dimension of such survivor sets as well as
results on certain bifurcation sets. Moreover, by taking unions of survivor
sets of intermediate $\beta$-transformations one obtains an important class of
sets arising in metric number theory, namely sets of badly approximable numbers
in non-integer bases. We prove, under the assumption that the underlying
symbolic space is of finite type, that these sets of badly approximable numbers
are winning in the sense of Schmidt games, and hence have the countable
intersection property, extending the results of Hu and Yu [2014], Tseng [2009]
and F\"{a}rm, Persson and Schmeling [2010].
