  Let $S= \{ a_{1}, a_{2}, \dots, a_{n} \}$ be a finite set of non-zero
integers. In \cite{KBAM21}, Karthick Babu and Anirban Mukhopadhyay calculated
the explicit structure of the Galois group of multi-quadratic field
$\mathbb{Q}(\sqrt{a_{1}}, \sqrt{a_{2}}, \dots, \sqrt{a_{n}})$ over
$\mathbb{Q}$. For a positive integer $d \geq 3$, $\zeta_{d}$ denotes the
primitive $d$-th root of unity. In this paper, we calculate the explicit
structure of the Galois group of $\mathbb{Q}(\sqrt{a_{1}}, \sqrt{a_{2}}, \dots,
\sqrt{a_{n}}, \zeta_{d})$ over $\mathbb{Q}$ in terms of its action on
$\zeta_{d}$ and $\sqrt{a_{i}}$ for $1 \leq i \leq n$.
