Translational tiles without spectra in finite abelian p-groups

We construct explicit translational tiles without spectra in three finite abelian $p$-groups. The first is a $64$-point subset of $\Z_4^4\times\Z_2^2$. The other two are a $512$-point subset of $\F_2^{13}$ and a $2187$-point subset of $\F_3^9$. Consequently, the tile-to-spectral implication fails fo