Duals of representations of affine group schemes
Let $R$ be a commutative ring. If $G$ is a group and $V$ is a dualizable i.e. finitely generated projective $R$-module on which $G$ acts, then it is known that $G$ also acts on the dual module $V^*$ via $(g \omega)(v)=\omega(g^{-1} v)$ and that $V^*$ becomes the dual in the tensor category of representations of $G$ over $R$.
Now I wonder if there is something similar for group schemes? (This should be a special case of Prop. 1.3.4 in Hovey's Homotopy theory for comodules over a Hopf algebroid.) So if $G$ is an affine group scheme over $R$ and $V$ is a dualizable $R$-module on which $G$ acts (i.e. we have natural group homomorphisms $G(A) \to \mathrm{Aut}_A(V \otimes_R A)$ for commutative $R$-algebras $A$), how to construct a dual representation $V^*$?
Here is my guess: Take the dual module $V^*$. There is a group anti-homomorphism $\mathrm{Aut}_A(V \otimes_R A) \to \mathrm{Aut}_A(V^* \otimes_R A)$ which maps a linear map to its dual. The inversion map is a group anti-homomorphism $G(A) \to G(A)$. Composition yields a group homomorphism $G(A) \to \mathrm{Aut}_A(V^* \otimes_R A)$. Can someone confirm this?
Specifically, how does this look like if $G=\mathrm{GL}_n$ and $V$ is the standard representation of rank $n$ (i.e. $R^n$ with obvious action)? The underlying $R$-module of $V^*$ will be $R^n$ (just as for $V$), but the action will be some non-trivial homomorphism $\mathrm{GL}_n(A) \to \mathrm{GL}_n(A)$ - which one? Out of curiosity, is there a classification of all homomorphisms of group schemes $\mathrm{GL}_n \to \mathrm{GL}_n$?