For a field $F$ we denote by $F^{\mathrm{ab}}$ the compositum of all finite Galois abelian extensions of $F$.

Is $\mathrm{Gal}(\mathbb{Q}_2(\sqrt[8]{3})^{\mathrm{ab}}/\mathbb{Q}_2(\sqrt[8]{3})) \cong \mathrm{Gal}(\mathbb{Q}_2(\sqrt[8]{48})^{\mathrm{ab}}/\mathbb{Q}_2(\sqrt[8]{48})) ? $

Equivalently,

Is every finite abelian group which is a Galois group of some finite Galois extension of $\mathbb{Q}_2(\sqrt[8]{3})$, is also a Galois group of some finite Galois extension of $\mathbb{Q}_2(\sqrt[8]{48})$ (and vice versa) ?