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.

What does $\mathrm{H}^1(k,\mathrm{PAut} X_{\bar k})$ classify?

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 $L$.

What does $\mathrm{H}^1(k,\mathrm{Aut}(X_{\bar k},L_{\bar k}))$ classify?