The poset of copies for automorphism groups of countable relational structures

Let $\mathrm{G}$ be a subgroup of the symmetric group $\mathfrak{S}(U)$ of all permutations of a countable set $U$. Let $\bar{\mathrm{G}}$ be the topological closure of $\mathrm{G}$ in the function topology on $U^U$. We initiate the study of the poset $\overline{$\mathrm{G}$}[U]:={f[U]\mid f\in \overline{$\mathrm{G}$}}$ofimagesofthefunctionsin$\overline{$\mathrm{G}$}$, being ordered under inclusion. This set $\bar{\mathrm{G}}[U]$ of subsets of the set $U$ will be called the \emph{poset of copies for} the group $\mathrm{G}$. A denomination being justified by the fact that for every subgroup $\mathrm{G}$ of the symmetric group $\mathfrak{S}(U)$ there exists a homogeneous relational structure $R$ on $U$ such that $\bar{G}$ is the set of embeddings of the homogeneous structure $R$ into itself and $\bar{\mathrm{G}}[U]$ is the set of copies of $R$ in $R$ and that the set of bijections $\bar{G}\cap \mathfrak{S}(U)$ of $U$ to $U$ forms the group of automorphisms of $\mathrm{R}$.