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Is there an embedding of $\mathrm{Aut}(M_{12})$ into the automorphism group of some larger sporadic group that fuses its two conjugacy classes of $\mathrm{PGL}(2,11)$ subgroups?

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Norton and Wilson - Maximal subgroups of the Harada–Norton group seems to imply that $\mathrm {HN} \rtimes C_2$ works, along with its overgroups $\mathbb{B}$ and $\mathbb{M}$.

Edit:… or not. This cannot be true, at least according to GAP, so I must’ve misunderstood the paper.

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