Lifting iterated Frobenius over $W_n(k)$

Let $X$ be a smooth scheme over a perfect field $k$ of positive characteristic. M.V.~Nori and V.~Srinivas studied infinitesimal liftings of Frobenius morphism $F:X\to X$. Namely, given a Frobenius lifting $F_Y:Y\to Y$ over $W_{n-1}(k)$ the obstruction of lifting $F_Y$ over $W_n(k)$ is a class in $H^1(X,T_X\text{tensor}{B}_1{\mathrm{\Omega }}_{X\mathrm{/}k}^1)$. We give a modified version of their theory and applications, in which we consider lifting $(n-1)$th iterated Frobenius $F^{n-1}:X\to X$ directly over $W_n(k)$.