Background: Recall from SGAI that a group $G$ acts admissibly on a scheme $X$ if the quotient $X \to X/G$ exists and is an affine morphism of schemes. This is the case if and only if every orbit of $G$ (meaning every set of the form $Gx$, with $x \in X$) is contained in an affine open subset of $X$. By graded prime avoidance it follows that this is the case for quasi-projective schemes, and more generally by EGA II Corollary 4.5.4 this is satisfied if $X$ has an ample invertible sheaf.
My question: Suppose that $X$ is a separated scheme of finite type over a field $k$ (not necessarily algebraically closed). Does any finite group $G$ act admissibly on $X$? Clearly this will be true if $X$ has a locally closed embedding into projective space, can we add some assumptions on $X$ that yields the aforementioned locally closed embedding?