An elementary exercise in group actions

Let $G$ be a group, $T$ a normal subgroup, and suppose that $G$ acts transitively on a set $X$. Then the induced action of $T$ on $X$ may not be transitive; let $T \setminus X$ denote the orbit space. Show that the quotient group $G/T$ acts well-definedly and transitively on $T \setminus X$.

 I plan to do this somewhat regularly... I know that I'm terrible about keeping up a store of problems and examples, which is a key component of mathematical pedagogy.