Apparently only Mathieu group $M_{1}2$ and Mathieu group $M_{2}4$.

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 groupsHmm, 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 k-transitive groups | 22, 155, 6
- K-transitive group | 19, 174, 7
- Mathieu group | 239, 427, 13
- Sporadic group | 8, 482, 17
- Classification of finite simple groups | 186, 949, 23
- Classification of finite groups | 69, 1k, 27
- Normal subgroup | 149, 1k, 30
- Quotient group | 51, 2k, 33
- Subgroup | 0, 2k, 35
- Group | 0, 5k, 89
- Algebra | 0, 8k, 171
- Mathematics | 17, 28k, 633
- Ciro Santilli's Homepage | 262, 218k, 4k