Extensions representing Nori-Srinivas obstruction

Let $(X,F)$ be a pair of a smooth variety X over an algebraically closed field k of characteristic $p>0$ and its Frobenius morphism F. Given a Frobenius $W_n\left(k\right)$-lifting $(\overline{X},\overline{F})$ of the pair $(X,F)$ for $n\ge 1$, Nori and Srinivas [9] determined the obstruction $ob{s}_{\overline{X},\overline{F}}\in \mathrm{Ext}({\Omega }_{X/k}^1,B{F}_{⁎}{\Omega }_{X/k}^1)$ to Frobenius $W_{n+1}\left(k\right)$-lifting of $(\overline{X},\overline{F})$ in terms of Čech cohomology. The extension representing $ob{s}_{\overline{X},\overline{F}}$ has been only known for $n=1$, which uses the Cartier operator. In this paper, we interpret $ob{s}_{\overline{X},\overline{F}}$ in terms of Kato's version of de Rham-Witt Cartier operator [8] and determine the extension representing $ob{s}_{\overline{X},\overline{F}}$ for $n\ge 2$.