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Recently we introduced a class $\widetilde{\mathcal{Z}}$ of $2$-local finitespectra 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 wegive an almost complete computation. The incompleteness lies in the fact thatwe are unable to eliminate one family of $d_3$-differentials and a fewpotential hidden extensions, though we conjecture that all these differentialsand hidden extensions are trivial.

Towards the $K(2)$-local homotopy groups of $Z$