  If $X$ is Frobenius split, then so is its normalization and we explore
conditions which imply the converse. To do this, we recall that given an
$\mathcal{O}_X$-linear map $\phi : F_* \mathcal{O}_X \to \mathcal{O}_X$, it
always extends to a map $\bar{\phi}$ on the normalization of $X$. In this
paper, we study when the surjectivity of $\bar{\phi}$ implies the surjectivity
of $\phi$. While this doesn't occur generally, we show it always happens if
certain tameness conditions are satisfied for the normalization map. Our result
has geometric consequences including a connection between $F$-pure
singularities and semi-log canonical singularities, and a more familiar version
of the ($F$-)inversion of adjunction formula.
