Let $G$ be a finite subgroup of $SL(n,\mathbb{C})$ and suppose that the only singular point of $\mathbb{C}^{n}/G$ is the origin. My question is the following: is there a $N>0$ s.t.there exist a $G$-invariant smooth map $i_{N}$
$$i_{N}:\mathbb{C}^{n}\rightarrow \mathbb{C}^{N}$$ and $$i_{N}:(\mathbb{C}^{n}\setminus\left\{0\right\})/G\rightarrow \mathbb{C}^{N}$$ is a smooth embedding? If it is not the case in general, are there conditions that guarantee the existence of such a map?

Does the situation change if the ambient group is $U(n)$ instead of $SL(n,\mathbb{C})$?

Let $G$ be a finite subgroup of $SL(n,\mathbb{C})$. Let $G$ act on $\mathbb{C}^{n}$ with the action induced by $SL(n,\mathbb{C})$ and the induced action of $G$ on the unit sphere $S^{2n-1}$ is free. My question is the following: is there a $N>0$ s.t.there exist a $G$-invariant holomorphic map $i_{N}$
$$i_{N}:\mathbb{C}^{n}\rightarrow \mathbb{C}^{N}$$ and $$i_{N}:(\mathbb{C}^{n}\setminus\left\{0\right\})/G\rightarrow \mathbb{C}^{N}$$ is a smooth embedding? If it is not the case in general, are there conditions that guarantee the existence of such a map?

Does the situation change if the ambient group is $U(n)$ instead of $SL(n,\mathbb{C})$?