Let $X$ be a smooth affine scheme over $\mathbb{Z}/{p^2}$ that is a complete intersection, say $X$ is the spectrum of $\mathbb{Z}/{p^2}[x_1,...x_n]/(f_1, ... f_r)$, where $n-r$ is the dimension of $X$. Let $X'$ be the corresponding scheme over $\mathbb{F}_p$ (just reduce everything mod $p$). On $X'$, we have the Frobenius endomorphism $F_{X'}$. I want to lift it to $X$, i.e. I want an endomorphism $F_X$ of $X$ such that it reduces to $F_{X'}$ mod $p$. We have the Frobenius on $\mathbb{A}^{n-r}_{\mathbb{F}_{p}}$ and hence a lift of this to $\mathbb{A}^{n-r}_{\mathbb{Z}/{p^2}}$ (this lift is not unique) and we have an étale morphism from $X$ to $\mathbb{A}^{n-r}_{\mathbb{Z}/{p^2}}$ by smoothness. Is there some way to get a lift of the Frobenius to $X$ using only the étale-ness of the latter map?
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