Let $G_{/\mathbf{Z}}$ be a Chevalley-Demazure group scheme, i.e. a split reductive group scheme over $\mathbf{Z}$. Let $\rho:G\to \operatorname{GL}(V_{/\mathbf{Z}})$ be a representation. If $k$ is a finite field, we get an induced representation $\rho(k):G(k)\to \operatorname{GL}(V\otimes k)$. In particular, we have a finite group $G(k)$ acting on a finite $k$-vector space $V_k$, so we can talk about the "ordinary" group cohomology $H(k)=\operatorname{H}^\bullet(G(k),V_k)$. Certainly, since everything is finite, one can in principle compute $H(k)$ for given $G$, $\rho$ and $k$. On the other hand, we could take cohomology $\operatorname{H}^\bullet(G,V)$ in the category of $G$-modules.
Is it the case that for the characteristic of $k$ sufficiently large (how large may depend on the root datum of $G$), we have $\operatorname{H}^\bullet(G,V)\otimes k \simeq H(k)$?
Either way, is there a way of computing $H(k)$, or at least $\dim H^i(k)$, that does not depend on $k$? (again, assuming the characteristic of $k$ is large enough)
Edit: I'm especially interested in the following special case. Let $G=\operatorname{PGL}(n)$ and $\rho:G\to \operatorname{GL}(\mathfrak{pgl}_n)$ be the adjoint representation. For a finite field $k$ (of sufficiently large characteristic relative to $n$) is it possible to compute $\operatorname{H}^1(\operatorname{PGL}_n(k),\mathfrak{pgl}_n(k))$?