  For $k$ a perfect field of characteristic $p>0$ and $G/k$ a split reductive
group with $p$ a non-torsion prime for $G,$ we compute the mod $p$ motivic
cohomology of the geometric classifying space $BG_{(r)}$, where $G_{(r)}$ is
the $r$th Frobenius kernel of $G.$ Our main tool is a motivic version of the
Eilenberg-Moore spectral sequence, due to Krishna.
  For a flat affine group scheme $G/k$ of finite type, we define a cycle class
map from the mod $p$ motivic cohomology of the classifying space $BG$ to the
mod $p$ \'etale motivic cohomology of the classifying stack $\mathcal{B}G.$
This also gives a cycle class map into the Hodge cohomology of $\mathcal{B}G.$
We study the cycle class map for some examples, including Frobenius kernels.
