There is a nice theorem, that states that for $G$ acting on $G$ (as group action on set) by product from the left- i.e for every $g_1,g_2 \in G, g_1\cdot g_2:=g_1g_2$; a stabilizer of $ s \in G$ , (which I'll notate by $G_s$ ), is normal if and only if it stabilise every element of $Gs$- as product of $s$ with the set from the left, pointwise.
I'm trying to come up with example that it is not always right: a group acts on set with not normal stabilizer for some element.
Can someone give some exapmle?