Automorphisms of Frobenius twisted de Rham cohomology

In this short paper, we prove that the moduli of automorphisms of Frobenius twisted de Rham cohomology functor is given by ${\mathbb{G}}_m$. Our method is to use the notion of ${\mathbb{G}}_a^{\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{f}}$-modules and its connection to the de Rham cohomology functor introduced in \cite{M22}. As an application of the induced ${\mathbb{G}}_m$-action, we reprove a result of Bhatt, Petrov, and Vologodsky on the decomposition of the Frobenius twisted de Rham complex.