Let $\operatorname{Sym}(X)$ denote the group of the bijections on the set $X$, and take $Y \subset X$. Is there any relation between the groups $\operatorname{Sym}(X)$ and $\operatorname{Sym}(Y)$?
Background. Let $R$ be a division ring; then, either $G:=(R,+)$ and $G^*:=(R \setminus \lbrace0\rbrace,\cdot)$ are groups (with, in particular, the former abelian). By Cayley Theorem, $G \cong \Theta \le \operatorname{Sym}(R)$ and $G^* \cong \Xi \le \operatorname{Sym}(R\setminus\lbrace0\rbrace)$, so that - before asking whether it makes sense to look for relations between $\Theta$ and $\Xi$ - I think we need to firstly address the same question for $\operatorname{Sym}(R)$ and $\operatorname{Sym}(R\setminus\lbrace0\rbrace)$ theirselves.
$\operatorname{Sym}(Y)$ embeds naturally as a subgroup in $\operatorname{Sym}(X)$ the following way: Take a bijection $f:Y\to Y$, and extend it to a bijection $f':X\to X$ by $f'(x) = x$ for any $x\in Y^c$. This subgroup is not normal in general.