Extensions of the Frobenius to the ring of differential operators on a polynomial algebra in prime characteristic

Let $K$ be a field of characteristic $p>0$. It is proved that each automorphism $\sigma \in \operatorname{Aut}_K(\mathcal D(P_n))$ of the ring $\mathcal D(P_n)$ of differential operators on a polynomial algebra $P_n= K[x_1, \ldots, x_n]$ is uniquely determined by the elements $\sigma (x_1), \ldots ,\sigma (x_n)$, and that the set $\operatorname{Frob}(\mathcal D(P_n))$ of all the extensions of the Frobenius (homomorphism) from certain maximal commutative polynomial subalgebras of $\mathcal D(P_n)$, such as $P_n$, to the ring $\mathcal D(P_n)$ is equal to $\operatorname{Aut}_K(\mathcal D(P_n) ) \cdot \mathcal{F}$ where $\mathcal{F}$ is the set of all the extensions of the Frobenius from $P_n$ to $\mathcal D(P_n)$ that leave invariant the subalgebra of scalar differential operators. The set $\mathcal{F}$ is found explicitly; it is large (a typical extension depends on countably many independent parameters).