Let M be a countable recursively saturated model of Th($\mathbb{N}$), and let G$=$Aut(M), considered as a topological group. We examine connections between initial segments of M and subgroups of G. In particular, for each of the following classes of subgroups H$<$G, we give characterizations of the class of terms of the topological group structure of H as a subgroup of G.

(a) $\{ H : H =G_{(K)}$ for some $K\prec_{\rm e} M \}$

(b) $\{ H : H =G_{\{K\}}$ for some $K\prec_{\rm e} M \}$

(c) $\{ H : H =G_{\{M(a)\}}$ for some $ a\in M \}$

(d) $\{ H : H =G_{\{M(a)\}}=G_a$ for some $ a\in M \}$

(Here, M(a) denotes the smallest $I\prec_{\rm e}$M containing a, $G_{\{A\}}=\{g \in G: A=\{gx: x\in A\} \}$, $G_{(A)}=\{g\in G: \forall a\in A\:ga=a\}$, and $G_a=\{g\in G: ga=a\}$.)