Model polynomials of $GF(p)$ as polynomials of $GF(p^n)$
Imagine that we have polynomials $$F_1, F_2, \ldots, F_n : \mathbb{F}_p \rightarrow \mathbb{F}_p\ ,$$ $n$ polynomials in $n$ variables, which define a map of sets $\mathbb{F}_q \rightarrow \mathbb{F}_q$, $q = p^n$, and we want to find the map as a single univariate polynomial over $\mathbb{F}_q$. This is a follow-up question that is answered already here for the reverse direction.
The reverse direction of the previous question would be taking $n$ polynomials in $n$ variables $F_1, F_2, \ldots, F_n : \mathbb{F}_p^n \rightarrow \mathbb{F}_p$ which (given a choice of generator for $\mathbb{F}_{p^n}$) define a map of sets $\mathbb{F}_{p^n} \to \mathbb{F}_{p^n}$, and trying to realize that map as a single univariate polynomial over $\mathbb{F}_{p^n}$.
Yes, I will update the question with the better description. But that's what I had in mind, is it possible?