Fundamental theorem of projective geometry states that given two projective frames $P=\{p_1,\ldots,p_{n+2}\}$ and $P'=\{p_1',\ldots, p_{n+2}'\}$ of a projective space $\mathbb{P}^{n}_k$, there is exactly one projective transformation $A\in PGL_{n+1}(k)$ that maps the first frame onto the second one.
Assume now that $K/k$ is a Galois extension with the Galois group $G$ and the sets $P$ and $P'$ (defined over $K$) are both $G$-invariant and have identical orbit structure w.r.t. $G$. Does there still exist $A\in PGL_{n+1}(k)$ (over small $k$) mapping $P$ to $P'$?