The Frobenius automorphism is special because the $p$-power map makes sense in any characteristic $p$ ring, which allows us to canonically extend the Galois-theoretic Frobenius to any such ring. I don't know of any other field automorphism having this property, and I wonder if there are any other examples.

In order to formalize my question, let $K$ be a field and $\sigma$ a nontrivial automorphism of $K$. Let $\mathcal C$ be the category of commutative, unital $K$-algebras. Given $R \in \mathcal C$, we can twist $R$ by $\sigma$ by pre-composing the structure map $K \to R$ with $\sigma$. The map $R\mapsto R_\sigma$ determines a functor $T_\sigma$ which is an automorphism of $\mathcal C$.

Let us say that $\sigma$ is *magical* if there exists a natural transformation $$M: \mathbf{1}_{\mathcal C} \to T_\sigma.$$

The data of $M$ is equivalent to the data of a $K$-algebra homomorphism $R \to R_\sigma$ for every $K$-algebra $R$, in a manner compatible with the morphisms in $\mathcal C$. By definition of the $K$-algebra structure on $R_\sigma$, this amounts to giving a *$\sigma$-linear* morphism $R \to R$, or, what is the same thing, *an extension of $\sigma$ to $R$*.

The following proposition illustrates that magical automorphisms are probably quite rare.

**Proposition 1:** Complex conjugation $\sigma \in \text{Gal}(\mathbf C/\mathbf R)$ is not magical.

I don't know if an easier proof is possible, but this one is fun:

*Proof*: I prove something stronger, namely that there exists a $\mathbf C$-algebra $R$ with no morphism $R\to R_\sigma$. Indeed, there exists an elliptic curve $E/\mathbf C$ such that $E$ is not isogenous to $E^\sigma$; any elliptic curve whose period lattice is not homothetic to a sublattice of its complex conjugate does the trick. Taking for $R$ the function field of $E$, we get a $\mathbf C$-algebra $R$ with no $\mathbf C$-algebra maps $R \to R_\sigma$.

*Remark*: In this case, giving an extension of $\sigma$ to $R$ could be viewed as putting an *almost real* structure on $R$. Indeed, any $\mathbf{C}$-algebra of the form $S \otimes_\mathbf{R} \mathbf{C}$, $S$ an $\mathbf{R}$-algebra, carries a natural extension of complex conjugation.

**Proposition 2:** If $K$ is a perfect field of characteristic $p$ and $\sigma$ is the $p$-power automorphism of $K$, then $\sigma^n$ is magical for every $n>0$.

*Proof*: For any $K$-algebra $R$, the $p^n$-th power endomorphism $R \to R$ is an extension of $\sigma^n$ to $R$, and it is obviously natural in $R$.

**Questions:** Does there exist an example of a magical automorphism which is not Frobenius? Are negative powers of Frobenius magical? Does there exist a magical automorphism in characteristic $0$? If $K/F$ is $\overline{\mathbf F_p}/\mathbf F_p$, and $\sigma \in G = \widehat{\mathbf Z}$ is not a positive integer power of Frobenius, is $\sigma$ magical? (My guess is that it is not.)

*Remark*: It seems that the fact that Frobenius is magical plays a basic role in mod $p$ and $p$-adic geometry, by giving rise to the notion of Frobenius morphism.