In this question, I am going to consider noncommutative projective algebraic geometry, as introduced by Artin and Zhang in the seminal paper Noncommutative projective schemes. The $\operatorname{Proj}$ construction which associaties to a graded commutative ring the scheme $\operatorname{Proj}(A)$ does not generalize to the noncommutative setting, because localization by multiplicative sets does generaly exsists and many noncommutative rings have very few prime ideals.
What does generalize is the category of coherent sheaves, which can be defined purely algebraically using certain module categories.
For the sake of simplicity, my base field will be algebraically closed of zero characteristic. For every finitely generated connected graded left and right Noetherian algebra $A$, we can consider the category $\operatorname{qgr}(A)$ which is the quotient of the category of finitely generated graded $A$-modules by its Serre subcategory of those modules of finite lenght. $\operatorname{qgr}(A)$ serves as an analogue of the category of choerent sheaves on $A$. The noncommutative projective scheme associated to $A$ is the triple $(\operatorname{qgr}(A), \mathcal{A}, s)$, where $\mathcal{A}$ is the image of $A$ seen as a module over itself in $\operatorname{qgr}(A)$, and $s$ is the autoequivalence of the category induced by the shift functor.
Classical (commutative) geometric invariant theory study quotients of algebraic varieties by actions of algebraic groups. In the general situation, there are many subleties that appear when we want to construct the quotients - and hence the field of geometric invariant theory. However, if our group $G$ is finite and the variety is quasi-projective, the definition of the quotient is not too hard.
Restricting attention to projective varieties and quotients by finite groups, I don't know how the geometric quotient reflects on the category of coherent sheaves (but I guess that this is known).
For the sake of simplicity, I am interested in the following situation. $A$ is a $\mathbb{N}$-graded algebra, with $A_0=k$, $\operatorname{dim} A_1 < \infty$ and $A$ generated in degree one. In particular $A$ is finitely generated and for every $n$, $\operatorname{dim}A_n < \infty$. I assume that the growth function $\gamma(n)=\operatorname{dim} A_n$ is polynomial, and hence the GK dimension of $A$ is finite and an integer. Finally I assume that $A$ is a left and right Noetherian domain. PS: I am aware that the might be some reduntancy in the conditions above.
Let $G$ be a finite group of graded automorphisms of $A$. $A^G$ again will be a left and right Noetherian domain and finitely generated, by Montgomery-Small generalization of classical Noether's theorem in invariant theory. Since $G$ acts by graded automorphisms, $A^G$ will again be $\mathbb{N}$-graded and, since $G$ is finite, have an integer Gelfand-Kirillov dimension (which, in fact, is the same as $\operatorname{GK}(A)$.) However, $A^G$ will in general not be generated in degree 1.
In the above situation, what should be the noncommutative geometric invarian theory quotient of $(\operatorname{qgr}(A), \mathcal{A}, s)$?
As I mentioned, my main interest is for noncommutative projective algebraic geometry. But, if for other theories of noncommutative algebraic geometry, there are interesting notions of a geometric quotient, I would also be glad to hear about them
