Let $X$ and $Y$ be complex projective varieties. Assume there is a finite group $G$ acting on $Y$ and we denote the quotient projective variety by $Y/G$. We have a morphism of $\mathcal{Hom}$-schemes induced by the quotient of the group action in the following form: $$f: \mathcal{Hom}(X,Y) \rightarrow \mathcal{Hom}(X, Y/G)$$ A morphism that can be lifted to the left side is called a liftable morphism. For a connected quasi-projective variety $U$, we say a morphism $\tilde{g}: U\rightarrow Y/G$ is a restriction of a morphism $g: X\rightarrow Y/G$ if there is an etale map $h:U\rightarrow X$ such $ \tilde{g}$ is the restriction of $g$ along $h$. Note that if we replace $X$ by the algebraic closure of its function field i.e. $\overline{K(X)}$, then all morphisms become liftable. This implies that every morphism on the right is liftable after some restriction.
- For every morphism on the right is there a neighborhood (with the topology on the $\mathcal{Hom}(X, Y/G)$ explained below) such that it becomes liftable after a restriction?
Edit: The topology on $\mathcal{Hom}(X, Y/G)$ can be considered to be the analytic topology of complex points on the $\text{Hom}$-scheme.
For the purpose of this problem you can assume $Y=(\mathbb{P}^d)^{\times n}$ for arbitrarily large $n$ and $d$, $G = S_n$ and $Y/G = \text{Sym}^{n}(\mathbb{P}^d)$
