Does there exist a quasisimple group $G$ and an odd prime $p$ such that $G$ has cyclic Sylow $p$-subgroups and a weakly real element of $p$-power order?


From [Strongly real elements of odd order in sporadic finite simple groups](https://mathoverflow.net/questions/224519/strongly-real-elements-of-odd-order-in-sporadic-finite-simple-groups) the only sporadic finite simple group which has weakly real $2$-regular elements is McL. But McL does not have cyclic Sylow $p$-subgroups for the prime orders $3$ and $5$ of the weakly real elements.



Quasisimple group with cyclic Sylow p-subgroup and weakly real p-elements?
${\rm SL}(2,q)$ for odd $q$ has a unique element of order $2$, which is central. So for any odd prime $p$ dividing $q-1$ or $q+1$ it has a cyclic Sylow $p$-subgroup $P$ and a generator of $P$ is conjugate to its inverse but not by an involution. The smallest example is ${\rm SL}(2,5)$ with $p=3$, and $p=5$ also works in this case.


There are other examples. Normalizers of cyclic Sylow $p$-subgroups acting irreducibly in symplectic groups seem to be a source of examples. The results I have are just from computer calculations, but they should not be hard to prove more generally. I would guess that you might find similar examples in other classical groups.


Specific examples are ${\rm Sp}(4,q)$ for $q=3,5,7$ with $p=5,13,5$ respectively, ${\rm Sp}(6,q)$ with $p=7$, and ${\rm Sp}(8,3)$ with $p=41$.


