  We continue the study of permutations of a finite regular semigroup that map
each element to one of its inverses, providing a complete description in the
case of semigroups whose idempotent generated subsemigroup is a union of
groups. We show, in two ways, how to construct an involution matching on the
semigroup of all transformations which either preserve or reverse orientation
of a finite cycle. Finally, by way of application, we prove that finite full
transformation semigroups have no cover by inverse semigroups that is closed
under intersection.
