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For the sporadic groups, I'm certain this must be known. Brauer--Fowler is a key starting point for the classification and I'm sure one starts with the knowledge of the centralisers of involutions in sporadic groups. I would expect this to also be in the volumes by GLS. Most of the information should also be contained in the ATLAS of finite groups.
For groups of lie type over a field of characteristic $p=2$ you're looking at a unipotent element, which is in bad characteristic for any group outside of type $\mathsf{A}$. However, the book by Liebeck--Seitz on unipotent conjugacy classes should contain the answer. The work by Mizuno on unipotent conjugacy classes in bad characteristic must also give this information.
I'm pretty sure that this list is known. For simple groups of Lie type defined over a field of odd characteristic the elements are semisimple and one can use the theory of algebraic groups to help you compute the centralisers. You'll find this information in §4.5 of "The Classification of the Finite Simple Groups: Number 3" by Gorenstein--Lyons--Solomon.
If you want to see how difficult this problem can be check out Geoff Robinsons DPhil Thesis. There he classifies the irreducible subgroups of $\mathrm{GL}_{11}(\mathbb{C})$. It's quite the feat considering it's pre-classification of finite simple groups. homepages.abdn.ac.uk/d.j.benson/papers/r/robinson/thesis.dvi
@FriederLadisch Thanks for the details. Apologies if my second comment wasn't clear. I was saying what the function does for a general group, i.e., possibly not a permutation group. In this case I think my comment is accurate. I should have made that clearer. I'd seen it invokes IsNaturalSymmetricGroup first, for permutation groups, but couldn't figure out exactly what that was doing. Thanks for finding where that function is defined.
As far as I can tell from the source its general method for an arbitrary group $G$ is as follows: see if $G$ is finite, see if $[G,G]$ is simple by determining normal closures of conjugacy classes, see if $G \cong \mathrm{A}_n$ with $n\in \{5,6\}$ by ad-hoc methods, now check if $|G| = n!/2$ for some $n\geqslant 7$, then by CFSG we have $G \cong \mathrm{A}_n$. Checkout the command 'IsomorphismTypeInfoFiniteSimpleGroup' in 'grp.gi'.
The beauty about GAP is that it's open source, so you can just look up the code. A quick grep of the source code shows the command is defined in the file 'grpnames.gi' in the lib folder. It seems to check whether the derived subgroup is an alternating group using the command 'IsAlternatingGroup', which is defined in the same file.
@JimHumphreys Ah, thanks! I missed that. I struggle to not read $p$ as the defining prime. Maybe, one could also say here that it's sufficient to find a $p$-block with trivial defect.
@PaulBroussous Thanks for clarifying. My comment was a reaction to reading this statement at the end of their article. I'll develop my comments into a proper answer.
For finite simple groups of Lie type it is in fact the case that every (ordinary) irreducible character occurs in the square of the Steinberg character, except for some special unitary groups. This was proved by Heide, Saxl, Tiep, and Zalesski - arxiv.org/abs/1209.1768.
Just in case anyone else falls into this trap, it is not true that an automorphism of $A$ lifts to an automorphism of $\mathbb{Z}^n$. Simply take $A = \mathbb{Z}/5\mathbb{Z}$ and let $\phi : A \to A$ be the automorphism defined by $\phi(1+5\mathbb{Z}) = 2+5\mathbb{Z}$. This does not lift to an automorphism of $\mathbb{Z}$ (but it does lift to an endomorphism). A complete solution to when such an automorphism lifts is given in the paper "Tame Automorphisms of Finitely Generated Abelian Groups" by Turner and Voce.
Jim, this is just to say that Digne--Michel do give a full proof of the existence of regular unipotent elements, at least over $\bar{\mathbb{F}}_p$. This is Theorem 14.13. They present Lusztig's argument showing that there are finitely many unipotent classes and then deduce the existence of regular unipotent elements from this. In fact, as is mentioned in the introduction to Lusztig's paper, if one knows that there are finitely many unipotent classes over $\bar{\mathbb{F}}_p$ then one knows this over any algebraically closed field $K$ of characteristic $p$. Thus their proof works over $K$.
Just to point out that this appears as Proposition 14.17 in Digne--Michel's book "Representations of Finite Groups of Lie Type". The argument they give is along the same lines as yours.
