  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 an
algorithm to describe the map $\pi^*
:\operatorname{CH}^*(G/B;\mathbb{F}_p)\longrightarrow
\operatorname{CH}^*(G;\mathbb{F}_p)$ in terms of Schubert cells. Taking
advantage 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 $\pi^*$ is
given for $E_6$ and $E_7$. Denote the group action map $\mu:G\times G/B\to
G/B$, we also give an explicit formula for $A$-type, $C$-type, $G_2$, $F_4$ of
the cohomology map $\mu^*: \operatorname{CH}^*(G/B;\mathbb{F}_p)\longrightarrow
\operatorname{CH}^*(G\times G/B;\mathbb{F}_p)$.
