  We prove that intermediate extensions of perverse constructible
$\mathbb{F}_p$-sheaves commute with smooth pullbacks for schemes admitting a
closed embedding into a smooth scheme over a field of characteristic $p$
(embeddable schemes for short). Along the way we also prove that the
equivalence of categories of Cartier crystals with unit $R[F]$-modules commutes
with $f^!$ for a smooth morphism $f: X \to Y$ of embeddable schemes.
