  Recently we introduced a class $\widetilde{\mathcal{Z}}$ of $2$-local finite
spectra and showed that all spectra $Z\in \widetilde{\mathcal{Z}}$ admit a
$v_2$-self-map of periodicity $1$. The aim of this article is to compute the
$K(2)$-local homotopy groups $\pi_*L_{K(2)}Z$ of all spectra $Z \in
\widetilde{\mathcal{Z}}$ using a homotopy fixed point spectral sequence, and we
give an almost complete computation. The incompleteness lies in the fact that
we are unable to eliminate one family of $d_3$-differentials and a few
potential hidden extensions, though we conjecture that all these differentials
and hidden extensions are trivial.
