  Let $(R,m)$ be a Noetherian local ring of characteristic $p>0$. We introduce
and study $F$-full and $F$-anti-nilpotent singularities, both are defined in
terms of the Frobenius actions on the local cohomology modules of $R$ supported
at the maximal ideal. We prove that if $R/(x)$ is $F$-full or
$F$-anti-nilpotent for a nonzerodivisor $x\in R$, then so is $R$. We use these
results to obtain new cases on the deformation of $F$-injectivity.
