Consider a totally ordered set $X$ with $n$ elements. Then we can canonically label its elements as $1, \dots, n$ with $1<2<\dots < n$. Now let $\sigma$ be a permutation (element of the symmetric group $S_n$). It determines a new order on $X$ by $\sigma(1) < \sigma(2) < \dots$.
Let $Y$ be a subset of $X$ with $p$ elements. Then restriction of the original as well as the new order on $X$ determine a total ordering on $Y$. These orders are then related by a uniquely determined permutation $\sigma_Y$ on $p$ elements.
Example: let $X=\{ 1, 2, 3\}$ and let $\sigma$ be the transposition of $2$ and $3$. Then $\sigma_{\{ 1 , 2 \}}$ is the neutral element of $S_2$, while $\sigma_{\{ 2 , 3 \}}$ is a transposition.
I am wondering whether this notion of restriction of permutations has some particular name. I suspect it might be a well-known and studied concept, but I have never encountered this.