The Main Conjecture for Imaginary quadratic fields for the split prime $p=2$

Let $\mathbb{K}$ be an imaginary quadratic field such that $2$ splits into two primes $\mathfrak{p}$ and $\bar{\mathfrak{p}}$. Let $\mathbb{K}_{\infty}$ be the unique $\mathbb{Z}_2$-extension of $\mathbb{K}$ unramified outside $\mathfrak{p}$. Let $\mathfrak{f}$ be an ideal coprime to $\mathfrak{p}$ and $\mathbb{L}$ be an arbitrary extension of $\mathbb{K}$ contained in the ray class field $\mathbb{K}(\mathfrak{p}^2\mathfrak{f})$. Let $\mathbb{L}_{\infty}=\mathbb{K}_{\infty}\mathbb{L}$ and let $\mathbb{M}$ be the maximal $p$-abelian, $\mathfrak{p}$-ramified extension of $\mathbb{L}_{\infty}$. We set $X=Gal(\mathbb{M}/\mathbb{L}_{\infty})$. In this paper we prove the Iwasawa main conjecture for the module $X$.