Let $G$ be a finite group and $Ω$ a transitive (hence finite), faithful $G$-set, and let the author of this question be someone who only has a superficial understanding of permutation group theory.
Trying to motivate the definition of a block system, I've noticed that $G$ acts on the set of all partitions of $Ω$, which we will denote by $\DeclareMathOperator{Part}{Part}\Part(Ω)$. I've noticed that block systems $ℬ\in \Part(Ω)$ precisely are the fixed points due to their $G$-invariance, i.e. $G_ℬ = G$.
Actually, I first noticed that the stabilizer of a block system is normal – so the following question seemed natural:
Question when is $G_Π$ a nontrivial normal subgroup?
I even struggled with the simpler question:
Question What is an example of $Π\in \Part(Ω)$ with $G_Π\unlhd G$?
Of course, this question becomes easy when we take $Π=\{\{x\}\mid x\in Ω\}$, which is stabilized only by $1$, so I'm asking for nontrivial stabilizers. I've added the “transitive” restriction because we can always take a product action $G\times G'\curvearrowright Ω\times Ω'$ where we get such a scenario “for free” by taking orbits of one component, respectively; but I find that horribly uninsightful.
I looked at easy permutation groups such as $S_3$ or $D_4$ ($Ω=\underline{4}$) and nontrivial partitions, but couldn't find an example.
I'm aware of this answer, which tells us that if $G_Π$ is normal, then what stabilizes $Π$ also stabilizes $gΠ$, so I guess the “more transitive” the action $G\curvearrowright \Part(Ω)$ is, the heavier this restriction is, and the less such partitions exist.