Picking the set of coset representatives of a stabilizer under a group action

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 $a$, $exists alphain U$ 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