Let $X$ be a projective variety over a perfect field $k$. Recall that a *twist* of $X$ is a variety $Y$ over $k$ such that $$X\_{\bar k} \cong Y\_{\bar k}.$$
The twists of $X$ are classified by the Galois cohomology set $\mathrm{H}^1(k,\mathrm{Aut} X\_{\bar k}).$ 


My question concerns what happens when one considers projective automorphisms instead. Namely, fix an embedding $X \subset \mathbb{P}^n$ and let $\mathrm{PAut} X$ denote the collection of automorphisms of $X$ which are induced by an automorphism of the ambient projective space.



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A first naive guess would be that this classifies varieties $Y \subset \mathbb{P}^n$ which become projectively isomorphic to $X$ over $\bar k$; however this is clearly seen to be false on taking $X = \mathbb{P}^n$.


It might help to put this problem into a more general framework. Namely, let $(X,L)$ be a projective variety equipped with a line bundle $L$. Denote by $\mathrm{Aut}(X,L)$ those automorphisms of $X$ which preserve the isomorphism class of $L$.



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Twists of projective automorphisms
Thanks to the hint from Ulrich, I think I am now able to answer the question.


The set $\mathrm{H}^1(k, \mathrm{Aut}(X\_{\bar k},L\_{\bar k}))$ classifes the following objects: 


$k$-Isomorphism classes of pairs $(Y,E)$, where $Y$ is a projective variety over $k$ and $E \in \mathrm{Pic}\_{Y/k}(k)$, which become isomorphic to $(X,L)$ after a finite separable field extension.


Here $\mathrm{Pic}\_{Y/k}$ denotes the Picard scheme of $Y$. Note that we have $$\mathrm{Pic}\_{Y/k}(k) = \mathrm{Pic}(\bar Y)^{\mathrm{Gal}(\bar k/ k)}.$$


