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

The $RO(C_{2^n})$-graded homotopy of $H\underline{\mathbb{Z}}$ through generalized Tate squares