In his paper 'Natural constructions of the Mathieu groups,' Curtis references an unpublished manuscript of G. Higman with a "significant constuction which makes use of the outer automorphism of $S_6$ to construct $M_{12},$ and the outer automorphism of $M_{12}$ to construct $M_{24}.$"

Does anyone know to what Curtis is referring? Is there another source for these constructions?