Let $\alpha_1$ be the first uncountable ordinal.
Let $U$ be the set of all $\alpha_1$-long enumerations of $\alpha_1$ (equivalently: $U$ is the set of bijections $\alpha_1 \to \alpha_1$).
Let $S\subseteq U$ be an unknown subset of $U$. Thus, $S$ is a set of sequences ... but we don't know what it is.
Given a countable subset $A\subseteq\alpha_1$, write $S|_A$ for the set of restrictions of elements of $S$, to $A$. Thus, $S|_A = \{ u|_A \mid u\in S\}$ where $u|_A$ is obtained in the natural way by restricting the uncountable sequence $u\in U$ to the indexes $A$ to obtain a countable subsequence $u|_A$.
Suppose we are given $$ F_S=\{ (A, S|_A) \mid \text{all countable }A\subseteq \alpha_1 \}. $$ Thus, we are given the function $F_S$ that if given a countable $A\subseteq\alpha_1$, will tell us the restriction of $S$ to $A$.
Can we reconstruct what $S$ was?
In general, no. For instance, let $\operatorname{id}$ be the identity bijection of $\alpha_1$. Then $F_U=F_{U\setminus \operatorname{id}}$, because each bijection between countable subsets of $\alpha_1$ can be extended to a non-identity bijection of $\alpha_1$. In particular, $U|_A= (U\setminus \operatorname{id})|_A $ for each countable subset $A$ of $\alpha_1$.