I am puzzled by the following structure (assume it is a group), called it $G$, contains and satisfies the following ingredients (in particular we can focus on the case that $d=2$). The $G$ contains generators including

$mathbb{Z}/2 equiv mathbb{Z}_2^A$

$SO(d)$

$mathbb{Z}/4 equiv mathbb{Z}_4^B$
The subindices $A$ and $B$ are to specify they are different groups. The group operations of $mathbb{Z}_2^A$, $SO(d)$, and $mathbb{Z}_4^B$ satisfy:
 (1) $mathbb{Z}_2^A$ and $SO(d)$ do not commute:
They obey $mathbb{Z}_2^A rtimes SO(d) = Spin(d)$, where $frac{Spin(d)}{mathbb{Z}_2^A}=SO(d)$. Namely, we have
$ 1 to mathbb{Z}_2^A to Spin(d) to SO(d) to 1$.
 (2) $SO(d)$ and $mathbb{Z}_4^B$ do not commute:
They obey $SO(d) rtimes mathbb{Z}_4^B =E(d)$, in fact $E(d)$ obeys that
the quotien group $frac{E(d)}{SO(d)}=mathbb{Z}_4^B$ and $frac{E(d)}{mathbb{Z}_2^B}=O(d)$
and $frac{mathbb{Z}_4^B}{mathbb{Z}_2^B}=mathbb{Z}_2’$ where $mathbb{Z}_2’$ is an another mod 2 abelian group. Namely, we have three short exact sequences
$ 1 to SO(d) to E(d) to mathbb{Z}_4^B to 1$, and
$ 1 to mathbb{Z}_2^B to E(d) to O(d) to 1$, and
$ 1 to mathbb{Z}_2^B to mathbb{Z}_4^B to mathbb{Z}_2′ to 1$.
We can also write $E(d)$ as
$$E(d)
={ (M, j) in (O(d), mathbb{Z}_4^B) ; vert ; det M = j^2}$$
where $j in mathbb{Z}_4^B$ here can be written as ${1,i,1,i}$ with $i^4=+1.$
 (3) $mathbb{Z}_4^B$ and $mathbb{Z}_2^A$ do not commute:
They obey $mathbb{Z}_4^B rtimes mathbb{Z}_2^A = D_8$ is a dihedral group of order 8. Namely, we have
$ 1 to mathbb{Z}_4^B to D_8 to mathbb{Z}_2^A to 1$.
In summary,
$G$ contain three group generators
$$
mathbb{Z}_2^A, SO(d),mathbb{Z}_4^B
$$
and group operations defined by
$$
mathbb{Z}_2^A rtimes SO(d) = Spin(d),;$$
$$
SO(d) rtimes mathbb{Z}_4^B =E(d),;$$
$$mathbb{Z}_4^B rtimes mathbb{Z}_2^A = D_8.
$$
The curious part is that the semi direct products $rtimes$ are given in a way that $ mathbb{Z}_2^A, SO(d),mathbb{Z}_4^B$ can be a normal subgroup repsectively in (1), (2), (3); but $ mathbb{Z}_2^A, SO(d),mathbb{Z}_4^B$ can be a quotient subgroup respectively in (3), (1), (2). So their group operations are cyclic in some way.
So what are the precise constructions/descriptions of the group $G$? Does this generate some familiar groups?
p.s. any comments/partial answers/Refs are welcome!