Let $\text{Sym}(U)$ be the symmetric group on a set $U$. Moreover, let $\Gamma$ by a subgroup of $\text{Sym}(U)$. From now on, we assume that $\Gamma$ acts transitively (in the natural way) on $U$. For a block $X \subseteq U$, we define a block system as follows $$\mathcal{X} = \{X^\gamma : \gamma \in \Gamma\}, \ \text{where } X^\gamma := \{x^\gamma : x \in X\}.$$
We consider also an action of $\Gamma$ on the block system $\mathcal{X}$ via $Y \mapsto Y^\gamma$. The stabilizer of $\mathcal{X}$ (wrt the action of $\Gamma$ on $\mathcal{X}$) is the set $$\Gamma_\mathcal{X} = \{\gamma \in \Gamma \ : \ \forall Y \in \mathcal{X} : Y^\gamma \in \mathcal{X}\}.$$
As far as I understand the definitions, I believe that $\Gamma = \Gamma_{\mathcal X}$ always hold, because there is no way to have have $Y^{\gamma} \notin \mathcal{X}$ for some $\gamma \in \Gamma$ and $Y \in \mathcal{X}$. The reason is as follows: We know that $Y = X^{\gamma'}$ for some $\gamma' \in \Gamma$, and so $Y^{\gamma} = X^{\gamma'\gamma}$. Since $\gamma\gamma' \in \Gamma$, we have $Y^{\gamma'} = X^{\gamma\gamma'} \in \mathcal X$ by the definition of $\mathcal{X}$.
However, the text I am reading (The Graph Isomorphism Problem by Martin Grohe) implicitly admits that this is not the case (and it is desirable that this is not the case). Could someone clarify what I might misunderstand or overlook?
The definition of $\Gamma_{\mathcal{X}}$ (given in my question) is the so called set-wise stabilizer of $\mathcal{X}$. One could also consider the point-wise stabilizer, which is defined as $$\Gamma_{(\mathcal{X})} = \{\gamma \in \Gamma : \forall Y \in \mathcal{X} : Y^\gamma = Y\}.$$
In the text (The Graph Isomorphism Problem by Martin Grohe), the author distinguishes between these notions by putting brackets around the set for which we are computing the stabilizer (i.e., $\Gamma_{\mathcal{X}}$ is a set-wise stabilizer and $\Gamma_{(\mathcal{X})}$ is a point-wise stabilizer).
However, specifically for block systems, there is later a redundant definition of stabilizer of a block system, which is defined as point-wise stabilizer but denoted as set-wise stabilizer (see pp 6.8 and pp. 6.25). This is where my confusion arose.
Thank you @DerekHolt for pointing me in the right direction.