On double covers of the generalized alternating group ${\displaystyle {\mathbb{Z}}_d{\wr }_m}$ as Galois groups over algebraic number fields

Let ${\displaystyle {\mathbb{Z}}_d\wr {\}}_m}$ be the generalized alternating group. We prove that all double covers of ${\displaystyle {\mathbb{Z}}_d\wr {\}}_m}$ occur as Galois groups over any algebraic number field. We further realize some of these double covers as the Galois groups of regular extensions of ℚ(T). If d is odd and m >7, then every central extension of ${\displaystyle {\mathbb{Z}}_d\wr {\}}_m}$ occurs as the Galois group of a regular extension of ℚ(T). We further improve some of our earlier results concerning double covers of the generalized symmetric group ${\displaystyle {\mathbb{Z}}_d{\wr }_m}$.