Is the category of finite flat group schemes equipped with "tensor products" and Hom-objects, encoding bilinear maps? I'm aware that the Cartier dual is $Hom(\mathbb{G}, \mathbb{G}_m)$, and want to know if this is part of a systematic collection of objects. For example, is there a "free ring scheme on $\mathbb{G}$"?

If so, given two affine group schemes whose underlying rings are free over the base, are there explicit descriptions of the tensor product and Hom objects in terms of the multiplication and comultiplication rules on the original rings?

Over a field, is there a description in terms of the Dieudonne correspondence?

(References, if they exist, would be very much appreciated. Thank you.)