Recall that an element of a finite group is said to be real if it is conjugate to its inverse, and strongly real if the conjugating element can be chosen to be an involution.

**Question:** Is it true that for the 26 sporadic finite simple groups, all real elements of odd order are strongly real, apart from elements in the *Atlas* classes 3A and 5A of the McLaughlin simple group $M^cL$?

My question is motivated by this Mathoverflow question of A.Rupinski: Why are there so few quaternionic representations of simple groups ? . As noted there, $M^cL$ is the only sporadic finite simple group which has `quaternionic' representations. From the *Atlas*, the irreducible characters $\chi_{10}$ and $\chi_{13}$ of $M^cL$ each have Frobenius-Schur indicator $-1$.

The literature on `strongly real' finite simple groups usually looks at all conjugacy classes, not the classes of odd order elements.

I suspect that the number of (irreducible) quaternionic representations of a finite group is greater than or equal to the number of real conjugacy classes of odd order elements which are not strongly real. This might even be a known open conjecture.

PS

General discussion on relation between numbers of real/complex/quaternionic conjugacy classes and irreducible representations can be found here: MO46900: Are there “real” vs. “quaternionic” conjugacy classes in finite groups?

If the Sylow 2-subgroups of $G$ are dihedral or large enough semi-dihedral, then a quaternionic representation exists iff a non-strongly real but real element of odd order exists. See his 1979 J.Algebra paper. (Although I imagine you know of this already...) $\endgroup$5more comments