  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 the generalized Tate squares. As an
example, we completely compute the $C_4$ case and investigate two
$\mathscr{P}$-homotopy limit spectral sequences for the family
$\mathscr{P}=\{e,C_2\}$.
