Quasisimple group with cyclic Sylow p-subgroup and weakly real p-elements? 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 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.
 A: ${\rm SL}(2,q)$ 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 $p=41$.
A: Regarding the question in your comment on Derek's answer, a 1978 paper of Bob Griess ( Quarterly Journal) proves that the only quasisimple group ( other than an ${\rm SL}(2,q))$ with all involutions central is the double cover of ${\rm A}_{7}.$
A Sylow $5$-subgroup of this double cover contains a weakly regular element of order $5$. This (with Derek's answer) seems to exhaust the quasisimple examples in which all involutions are central. But of course there might ( a priori at least) be weakly regular $p$-elements in other quasisimple groups where the involutions are not all central.
