  We explore the singularity classes $F$-nilpotent, weakly $F$-nilpotent, and
generalized weakly $F$-nilpotent under faithfully flat local ring maps. As an
application, we show that the loci of primes in a Noetherian ring of prime
characteristic which define either weakly $F$-nilpotent or $F$-nilpotent local
rings are open with respect to the Zariski topology whenever $R$ is $F$-finite
or essentially of finite type over an excellent local ring.

Commutative Algebra