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 ${\mathbb{K}}_{\mathrm{\infty }}$ 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 ${\mathbb{L}}_{\mathrm{\infty }}={\mathbb{K}}_{\mathrm{\infty }}\mathbb{L}$ and let $\mathbb{M}$ be the maximal $p$-abelian, $\mathfrak{p}$-ramified extension of ${\mathbb{L}}_{\mathrm{\infty }}$. We set $X=Gal(\mathbb{M}\mathrm{/}{\mathbb{L}}_{\mathrm{\infty }})$. In this paper we prove the Iwasawa main conjecture for the module $X$.