**The background**: We recall/define the following:

- $\Omega_n=\{1,\dots,n\}$.
- $M_n$ is the Mathieu group of degree $n$. We follow the Wikipedia article "Mathieu group" and define these groups for each of the 10 $n$ values 8-12 and 20-24. In each of these two series, $M_{k-1}$ is the point stabilizer in $M_k$ of $k\in\Omega_k$.
- Given an action of a group $G$ on a set $\Omega$ with a subset $\Delta\subseteq\Omega$, ${\mathbf C}_G(\Delta)$ denotes the pointwise stabilizer of $\Delta$ in $G$ (with set braces omitted when $\Delta$ is a singleton $\{\delta\}$).
- An action (of $G$ on $\Omega$) is
*half-transitive*when either (a) $|G|>1$ and every orbit has the same nontrivial cardinality or (b) [the degenerate case] $|G|=|\Omega|=1$.

**The question**: Given the Mathieu group $M_{20}$ as a permutation group on $\Omega_{20}$, is the induced action of ${\mathbf C}_{M_{20}}(20)$ on $\Omega_{19}$ half-transitive?

**The motivation**: Just like transitivity and primitivity, half-transitivity has higher-fold gradations $(n+\frac 12)$-transitivity for each integer $n\geq 1$. Indeed, like the way $n$-transitivity and $n$-primitivity interleave, it is true that $(n+\frac 12)$-transitive implies $n$-transitive, which then implies $(n-\frac12)$-transitive. It is also known that an $n$-transitive action is $(n+\frac 12)$-transitive if and only if for each $\Delta\subseteq\Omega$ with $|\Delta|=n$, the induced action of ${\mathbf C}_G(\Delta)$ on $\Omega\setminus\Delta$ is half-transitive. The parallel statement involving a reduction by one on $n$ holds when points $\delta\in\Omega$ replace the cardinality $n$ subsets $\Delta\subseteq\Omega$.

In particular, these lead to the following statements being equivalent; either they are all true or they are all false.

- The action of $M_{24}$ on $\Omega_{24}$ is $\frac{11}2$-transitive.
- The action of $M_{23}$ on $\Omega_{23}$ is $\frac 92$-transitive.
- The action of $M_{22}$ on $\Omega_{22}$ is $\frac 72$-transitive.
- The action of $M_{21}$ on $\Omega_{21}$ is $\frac 52$-transitive.
- The action of $M_{20}$ on $\Omega_{20}$ is $\frac 32$-transitive.
- The action of ${\mathbf C}_{M_{20}}(20)$ on $\Omega_{19}$ is $\frac 12$-transitive.

My personal curiosity revolves around the $4$- and $5$-transitive actions interacting with higher-fold half-transitivity at the upper end of this list.

The Wikipedia article goes on to say that $M_{20}$ has the form $2^4:A_5$, but I have no idea what action this entails which leaves as unknown the form of the point stabilizer in this group.

As an aside, the corresponding question applied to the lower 5 Mathieu groups $M_8$ through $M_{12}$ similarly reduces to whether or not ${\mathbf C}_{M_8}(8)$ is half-transitive on $\Omega_7$. In that case, $M_8$ acts regularly on $\Omega_8$ (indeed, the action is the regular action of the quaternion group on itself), guaranteeing ${\mathbf C}_{M_8}(8)$ is trivial. As $|\Omega_7|=7>1$ is not trivial, this action is not half-transitive.

**EDIT** (9 Sep 2022) I previously commented on not knowing what the action of $A_5$ on $2^4$ was. I now understand it to be $SL_2(\mathbb F_4)\cong A_5$ in its natural action on $\mathbb F_4{\!}^2$.