  Let $G$ be a reductive group over $\mathbb{F}_{p}$ with associated finite
group of Lie type $G^{F}$. Let $T$ be a maximal torus contained inside a Borel
$B$ of $G$. We relate the (rational) Tate motives of $\text{B}G^{F}$ with the
$T$-equivariant Tate motives of the flag variety $G/B$. On the way, we show
that for a reductive group $G$ over a field $k$, with maximal Torus $T$ and
absolute Weyl group $W$, acting on a smooth finite type $k$-scheme $X$, we have
an isomorphism $A^{n}_{G}(X,m)_{\mathbb{Q}}\cong
A^{n}_{T}(X,m)_{\mathbb{Q}}^{W}$ extending the classical result of
Edidin-Graham to higher equivariant Chow groups in the non-split case. We also
extend our main result to reductive group schemes over a regular base that
admit maximal tori. Further, we apply our methods to more general quotient
stacks. In this way, we are able to compute the motive of the stack of $G$-zips
introduced by Pink-Wedhorn-Ziegler for reductive groups over fields of positive
characteristic.
