Extending real-valued characters of finite general linear and unitary groups on elements related to regular unipotents

When n is odd, consider the finite general linear and unitary groups of rank n, extended by the inverse transpose automorphism. There are elements in the extended groups which square to a regular unipotent element, and we evaluate the values of irreducible characters of the extended groups on these elements. Several intermediate results on real conjugacy classes and real-valued characters of these groups are obtained along the way.