  The study of Frobenius actions on local cohomology modules over a local ring
of prime characteristic has interesting connections with the theory of tight
closure. This paper establishes new connections by developing the notion of
relative Frobenius actions on local cohomology. As an application, we show that
a ring has $F$-nilpotent singularities if and only if the tight closure of
every parameter ideal is equal to its Frobenius closure.
