Cartier crystals and perverse constructible \'etale $p$-torsion sheaves

For an $F$-finite scheme $X$ separated over a perfect field $k$ of characteristic $p>0$ which admits an embedding into a smooth $k$-scheme, we establish an equivalence between the bounded derived categories of Cartier crystals on $X$ and constructible $\mathbb{Z}/p\mathbb{Z}$-sheaves on the \'{e}tale site $X_{\text{\'{e}t}}$. The key intermediate step is to extend the category of locally finitely generated unit $\mathcal{O}_{F,X}$-modules for smooth schemes introduced by Emerton and Kisin to embeddable schemes. On the one hand, this category is equivalent to Cartier crystals. On the other hand, by using Emerton-Kisin's Riemann-Hilbert correspondence, we show that it is equivalent to Gabber's category of perverse sheaves in $D_c^b(X_{\text{\'{e}t}},\mathbb{Z}/p\mathbb{Z})$. Furthermore, we define intermediate extensions for Cartier crystals and show that our equivalence between Cartier crystals and perverse constructible \'{e}tale sheaves commutes with the intermediate extension functor.