My question is as follows:
Suppose that $G$ is a finite group that acts on finite set $A$.
Let $a \in A$.
Let $U$ be a set of left coset representatives of $Stab_{a}$.
Can $U$ be chosen such that for all $b$ in the orbit of $a$, $\exists \alpha\in U$ such that $\alpha b=a$?
It is easy to see that this is always possible if $Stab_{a}$ is normal since we can get the quotient group but the general result is less clear whether or not it is true.
edit: left coset rep