TITLE: Strongly real elements of odd order in sporadic finite simple groups
QUESTION [9 upvotes]: 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?

REPLY [4 votes]: One more confirmation of OP's observation can be found in tables,
from the paper:
Jordan Journal of Mathematics and Statisticscs (JJMS) l(2), 2008, pp. 97-103
97
STRONGLY REAL ELEMENTS IN
SPORADIC GROUPS AND ALTERNATING GROUPS, 
IBRAHIM SULEIMAN
One can observe many real, but not strongly real classes, but most of them
have even order, except group McL classes 3A, 5A - exactly as OP's proposal.
Table 3 will give the real classes which are not strongly real in Sporadic Groups
M11 all real elements are strongly real.
M12 all real elements are strongly real
M22 8A.
M23 8A
HS all real elements are strongly real.
J3 all real elements are strongly real.
M24 all real elements are strongly real.
McL 3A, 5A, 6A.
He all real elements are strongly real.
Ru all real elements are strongly real. 
Suz all real elements are strongly real.
ON all real elements are strongly real.
Co3 all real elements are strongly real.
Co2 16B.
Fi22 all real elements are strongly real.
HN 8A .
Ly all real elements are strongly real.
Th 8B.
Fi23 16AB, 22BC, 23AB.
Co1 all real elements are strongly real.
J4 all real elements are strongly real.
Fi_24 all real elements are strongly real.
BM all real elements are strongly real.
M 8C, 8F, 24F, 24G, 24H, 24J, 32A, 32B, 40A, 48A. 

Table 4 will give the complete list for all non real elements in the Sporadic Groups
M11 8AB, 11AB.
M12 11AB.
J1 None
M22 7AB.
J2 None
M23 7AB, 11AB, 14AB, 15AB, 23AB.
HS 14AB.
J3 19AB.
M24 7AB, 14AB, 15AB, 21AB, 23AB.
McL 7AB, 9AB, 11AB, 14AB, 15AB, 30AB.
He 7AB, 7DE, 14AB, 14CD, 21CD, 28AB.
Ru 16AB.
ON 31AB.
Co3 11AB, 22AB, 20AB, 23AB.
Co2 14AB, 15BC, 23BC, 30BC.
Fi22 11AB, 16AB, 18AB, 22AB.
HN 19AB, 35AB, 40AB.
Ly 11AB, 22AB, 33AB. 
Fi23 16AB, 22BC, 23AB.
Co1 23AB, 39AB.
J4 7AB, 14AB, 14CD, 21AB, 28AB, 35AB, 42AB.
Fi_24 18GH, 23AB.
BM 23AB, 30GH, 31AB, 32CD, 46AB, 47AB.
 M 23AB, 31AB, 39CD, 40CD, 44AB, 46AB, 46CD, 47AB, 56BC, 59AB,
62AB, 69AB, 71AB, 78BC, 87AB, 88AB, 92AB, 93AB, 94AB, 95AB,
104AB, 119AB.