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

Intermediate extensions of perverse constructible $\mathbb{F}_p$-sheaves commute with smooth pullbacks