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).



Fixed points of the automorphisms of sporadic groups
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 $K$.


