I am thinking about the situation if a group $G$ acts on a set $X=\{1,2,3,4,5\}$ by left multiplication where $\{1,2,3,4,5\}$ are cosets, so that we have a homomorphism $\phi:G \to S_5$. Suppose there is a $g\in G$ such that $\phi(g)$ will move $4$ to $5$. What is the group action $\phi'$ of $G$ on $X'=\{1,2,3,4\}$ by left multiplication?
I know $\phi'$ is a homomorphism between $G$ and $S_4$. However, the same $g$ under the group action by left multiplication will move $4$ to $5$, but $5$ is not in $X'$.