@Jason : Do you know any references dealing with such kind of questions? I could not find any. I'm still not getting the whole picture, I think. For example I would be happy to understand what happens if instead of a birational map, I just look at blow ups. If the class $\xi$ ramifies in just one irreducible divisor $D$ and one blows up a smooth $Z\subset D$, the ramification of $f^{*}(\xi)$ lies in the union of the strict transform of $D$ and the exceptional divisor. When is the exceptional divisor ramified? Can this be understood in terms of $Z$? Can the strict transform be unramified?

Have you had a look at "Geometry of the moduli spaces of sheaves" by Huybrechts and Lehn? They construct these coarse moduli spaces for polarized projective schemes. They also construct a relative version: a coarse moduli scheme for sheaves on the fibers of a projective morphisms with a relative ample line bundle. Maybe this is a good starting point, see ncatlab.org/nlab/files/HuybrechtsLehn.pdf

I don't see how you get to your translation.But isn't the line bundle $O_{\mathbb{P}^1_C}(Z\times C)$ just the pullback of $O_{\mathbb{P}^1_k}(1)$ via the other projection $\pi_{\mathbb{P}^1_k}$?

@Jason Starr: Thanks. I think i can mainly follow. But what associated Severi-Brauer variety do you mean? Shall we pick an order $O$ (maximal?) which has $\xi$ as its Brauer class at the generic point of $\mathbb{P}^n$ and look at $SB(O)\rightarrow \mathbb{P}^n$? The existence of a rational section implies that the Sveri Brauer variety is the projectivization of a locally free sheaf, so the algebra must be unramified?

Thanks, I wasn't aware of these trinagle identities. Should refresh my categroy theory.

Thanks, i never thought of using Cech cohomology. Nice and easy.

Thank you both. The hint that such $G$ are of the form $f^{*}H$ was what I needed. This solves my problem completely. If someone will write a short answer, then I can accept it.