Apparently only <Mathieu group M_{12}> and <Mathieu group M_{24}>.

http://www.maths.qmul.ac.uk/~pjc/pps/pps9.pdf mentions:
> The automorphism group of the extended Golay code is the 54-transitive Mathieu group $M_{24}$. This is one of only two finite 5-transitive groups other than symmetric and alternating groups
Hmm, is that 54, or more likely 5 and 4?

https://scite.ai/reports/4-homogeneous-groups-EAKY21 quotes https://link.springer.com/article/10.1007%2FBF01111290 which suggests that is is also another one of the Mathieu groups, https://math.stackexchange.com/questions/698327/classification-of-triply-transitive-finite-groups#comment7650505_3721840 and https://en.wikipedia.org/wiki/Mathieu_group_M12 mentions $M_{12}$.

Classification of 5-transitive groups

Mathieu group M12​

Ciro Santilli @cirosantilli 35

 Incoming links: Mathieu group $M_{12}$

Ciro Santilli @cirosantilli 35

 Incoming links: Mathieu group M12​

 Incoming links: Mathieu group $M_{12}$