We propose a new method to compute the \(C_{2^n}\) -equivariant homotopy groups of the Eilenberg–Mac Lane spectrum \(H\underline{\mathbb {Z}}\) as a \(RO(C_{2^n})\) -graded Green functor using generalized Tate squares. As an example, we completely compute the \(C_4\) -equivariant homotopy group of \(H\underline{\mathbb {Z}}\) as a Green functor. In the process, we investigate two \(\mathscr {P}\) -homotopy limit spectral sequences for the group \(C_4\) and the family \(\mathscr {P}=\{e,C_2\}\) .
