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Let $G$ be a connected reductive group, and $G/B$ be its flag variety. Let$\pi:G\to G/B$ be the natural projection. In this paper, we developed analgorithm to describe the map $\pi^*:\operatorname{CH}^*(G/B;\mathbb{F}_p)\longrightarrow\operatorname{CH}^*(G;\mathbb{F}_p)$ in terms of Schubert cells. Takingadvantage of the Pieri rule, we give an explicit formula for $A$-type,$C$-type, $G_2$, $F_4$ of the cohomology map $\pi^*:\operatorname{CH}^*(G/B;\mathbb{F}_p)\longrightarrow\operatorname{CH}^*(G;\mathbb{F}_p)$, and some partial result of $\pi^*$ isgiven for $E_6$ and $E_7$. Denote the group action map $\mu:G\times G/B\toG/B$, we also give an explicit formula for $A$-type, $C$-type, $G_2$, $F_4$ ofthe cohomology map $\mu^*: \operatorname{CH}^*(G/B;\mathbb{F}_p)\longrightarrow\operatorname{CH}^*(G\times G/B;\mathbb{F}_p)$.

Computations of the Comodule Structures of the Chow rings of Flag Varieties