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Sporadic groups have very few outer automorphisms (in fact, $|\mathrm{Out}(G)|\leqslant2$), so it is very natural to ask what are the fixed points subgroups. For a group of Lie type (and a suitable outer automorphism) one can get a twisted group or the same group over a subfield, but what about sporadic groups? I consulted several sourced but could not find the answer (or, perhaps, I am just not fluent enough in the language of the Atlas).

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    $\begingroup$ The ATLAS lists the orders of the centralizers in $G$ of all elements in ${\rm Out}(G)$, which provides you with some information. For all but the largest of the sporadic groups, you could compute these subgroups in GAP or Magma. I am not really sure exactly what you want to know about them - perhaps you need to ask a more specific question. $\endgroup$ – Derek Holt Jan 15 at 14:37
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I think the information you are looking for is in Table~5.3 of GLS3:

Gorenstein, Daniel; Lyons, Richard; Solomon, Ronald. The classification of the finite simple groups. Number 3. Part I. Chapter A. American Mathematical Society, Providence, RI, 1998. xvi+419 pp. ISBN: 0-8218-0391-3

For $K$ a sporadic group, there you will find an enumeration of the conjugacy classes of subgroups $\langle x \rangle$ of prime order in $\operatorname{Aut}(K)$, together with descriptions of the structure of $N_K(\langle x \rangle)$. So when $x$ is noninner, this gives exactly the fixed subgroup in $K$.

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