Let $G$ be a finite group scheme over the field $k$ ( you can assume $k$ is algebraically closed). I've seen the term "inner automorphisms" for $G$ at many places but I still don't understand what the correct definition. Suppose, we have a automorphism $f:G \to G$. This is means we have an automorphism $f_R:G(R) \to G(R)$ for every $k$-algebra $R$. So, $f$ is an inner-automorphism if $F_R$ is an inner-automorphism for every $k$-algebra $R$?
Also, is there a simpler definition? Or simpler way to think about it? Is it enough to say just $f_k$ is an inner-automorphism?
Also, any kind of reference on this topic would be very much appreciated.
Note that for any $g$ in $G(k)$ one has a morphism $c_g:G\to G$ given on $R$-points by $c_g(x)=gxg^{-1}$ which makes sense since $g\in G(k)\subseteq G(R)$. An automorphism of the form $c_g$ is called a naive inner automorphism of . Of course, if you replace $k$ by a ring nothing changes.
If $k$ is algebraically closed one then defines an automorphism to be inner if it is of the form $c_g$ for $g\in G(k)$ (in other words if it is naively inner). If you now assume that $k$ is not necessarily algebraically closed things get slightly more complicated. Namely, note that the association $G\to \mathrm{Aut}(G)$ which on $R$-points associates $g\in G(R)$ to the naive inner automorphism $c_g\in \mathrm{Aut}(G_R)$ certainly has kernel equal to $Z(G)$. Thus, we get an induced injection $G^\mathrm{ad}:=G/Z(G)\hookrightarrow \mathrm{Aut}(G)$. One then defines an automorphism of $G$ to be inner if its in the image of the map $G^\mathrm{ad}(k)\to \mathrm{Aut}(G)$. If $k$ is not algebraically closed then this is strictly larger than the set $c_g$ for $g\in G(k)$ since the map $G(k)\to G^\mathrm{ad}(k)$ needn't be surjective (e.g. think about $G=\mathrm{GL}_n$!). Intuitively, the inner automorphisms of $G$ are then those of the form $c_g$ for $g\in G(\overline{k})$ such that $c_g$ stabilizes $G(k)\subseteq G(\overline{k})$.
As an example of this phenomenon, one can note that $\begin{pmatrix}0 & i\\ i & 0\end{pmatrix}$ is an element of $\mathrm{PSL}_2(\mathbb{Q})$ and so gives an automorphism of $\text{SL}_{2,\mathbb{Q}}$ which is inner, but is not of the form $c_g$ for $g\in \mathrm{SL}_2(\mathbb{Q})$. Namely, note that
$$\begin{pmatrix}0 & i\\ i & 0\end{pmatrix}\begin{pmatrix}a & b\\ c & d\end{pmatrix}\begin{pmatrix}0 & i\\ i & 0\end{pmatrix}^{-1}=\begin{pmatrix}d & c\\ b & a\end{pmatrix}$$