> **Definition.** An action of a group $G$ on a set $X$ is *strongly $n$-transitive* if $G$ acts transitively on $n$-tuples of distinct elements in $X$ (via the diagonal action), and is *$n$-transitive* if $G$ acts transitively on *unordered* subsets of size $n$ in $X$.

The automorphism group of a connected topological manifold of dimension at least two acts strongly $n$-transitively for all $n$. On the other hand, for actions of groups that are smaller, the situation is different:

> **Theorem.** (Kramer, following Tits) There is no locally compact, $\sigma$-compact topological transformation group $G$ which acts effectively and $4$-transitively on a Hausdorff non-totally-disconnected space $X$. 

(As a side note, by stating the theorem like this, I am butchering a beautiful and intricate classification of all $2$-transitive group actions satisfying the other hypotheses) 

My question is whether there is any room between these extremes by weakening Kramer's hypotheses:

> **Question.** Does there exist a group $G$ that acts (strongly?) $n$-transitively but not $(n+1)$-transitively on a space $X$ for $n>3$?

For finite discrete or indiscrete $X$, the symmetric and alternating groups are examples, along with some of the Mathieu groups, but let's exclude those. My preference would be for either $X$ or $X/G$ to be a connected manifold but I'll take what I can get. 

This question was inspired in part by [this question][1].

<cite authors="Linus Kramer" mrnumber="2009240" cite="_J. Reine Angew. Math._ **563** (2003), 83--113">_Linus Kramer_, MR 2009240 [**Two-transitive Lie groups**](http://dx.doi.org/10.1515/crll.2003.085), _J. Reine Angew. Math._ **563** (2003), 83--113.</cite>


  [1]: http://mathoverflow.net/questions/245312/kappa-homogeneous-topological-spaces