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

Previously (Adv. Math. 360 (2020) art. id. 106895), we introduced a class (Z) over tilde of 2-local finite spectra and showed that all spectra Z is an element of (Z) over tilde admit a v(2)-self-map of periodicity 1. The aim here is to compute the K(2)-local homotopy groups pi(*)L(K(2))Z of all spectra Z is an element of (Z) over tilde using a homotopy fixed point spectral sequence, and we give an almost complete answer. The incompleteness lies in the fact that we are unable to eliminate one family of d(3)-differentials and a few potential hidden 2-extensions, though we conjecture that all these differentials and hidden extensions are trivial.