  We use tools of representation theory to get a better understanding of the
cohomology of graded group schemes. For that, we focus our attention on the
case in which the base field is of characteristic $p > 0$. Using as inspiration
the work of Friedlander, et al, we build the theory of graded $1$-parameter
subgroups denoted by $V_r^*(G)$. We give a natural homomorphism of bigraded
$\boldsymbol{\rm k}$-algebras $\psi: H^{*, *}(G, \boldsymbol{\rm k}) \to
\boldsymbol{\rm k}[V_r^* (G)],$ where $\boldsymbol{\rm k}[V_r^* (G)]$ is the
bigraded coordinate ring for $V_r^*(G)$. We show that $\psi$ is an
$F$-monomorphism for a class of graded group schemes. This provides evidence
that with the appropriate detection property, a Quillen-type result could exist
for graded group schemes.
