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I am looking for any modern (and English) reference to the following theorem:

If $G$ is finite group acting sharply $3$-transitively on $1+2^m$ points, then $G \cong {\mathrm{PSL}}(2,2^m)$.

I have looked at Huppert, Endliche Gruppen I, which seems to discuss the aspects on the converse direction (at least until chapter II) unless I am mistaken (my German is not too good). William Kerby's lecture note on "On infinite sharply multiply transitive groups" has it (Thm.10.2), however uses abstractions through skew fields.

I was wondering if there is a self contained proof without too much of details.

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