The refined local lifting problem for cyclic covers of order four

Suppose $\phi$ is a $\mathbb{Z}/4$-cover of a curve over an algebraically closed field $k$ of characteristic $2$, and $\Phi_1$ is a \emph{nice} lift of $\phi$'s $\mathbb{Z}/2$-sub-cover to a complete discrete valuation ring $R$ in characteristic zero. We show that there exist a finite extension $R'$ of $R$, which is determined by $\Phi_1$, and a lift $\Phi$ of $\phi$ to $R'$ whose $\mathbb{Z}/2$-sub-cover isomorphic to $\Phi_1 \otimes_R R'$. That result gives a non-trivial family of cyclic covers where Sa{\"i}di's refined lifting conjecture holds. In addition, the manuscript exhibits some phenomena that may shed some light on the mysterious moduli space of wildly ramified Galois covers.