Spectral eigenmatrix of the planar spectral measures with four elements

We consider the spectral eigenmatrix problem of the planar self-similar spectral measures μ_Q,D generated by

$$Q = \left({\matrix{{2q} & 0 \cr 0 & {2q} \cr}} \right)\,\,\,{\rm{and}}\,\,\,D = \left\{{\left({\matrix{0 \cr 0 \cr}} \right),\left({\matrix{1 \cr 0 \cr}} \right),\left({\matrix{0 \cr 1 \cr}} \right),\left({\matrix{{- 1} \cr {- 1} \cr}} \right)} \right\},$$

where q ≥ 2 is an integer. For matrix R ∈ M₂(ℤ), we prove that there exist some spectrum Λ such that Λ and RΛ are both the spectra of μ_Q,D if and only if det R ∈ 2ℤ + 1.