Let $(R,\mathfrak m)$ be a Noetherian local ring of prime characteristic $p$. For integer $e>0$, let $F^e_* R$ denote the $R$-module which is $R$ as an abelian group, but the $R$-module structure is given by $r\cdot s=r^{p^e}s, \forall r \in R, s\in F^e_* R$. Assume that $F^1_* R$ is a finitely generated $R$-module (hence so is $F^e_* R$ for all $e>0$).
My question is: Must it be necessarily true that $R$ is a direct summand of $F^e_*R$ for some $e>0$?