TITLE: Ideals invariant under ring automorphisms
QUESTION [7 upvotes]: I am looking for ideals $I\subset \mathbb{F}_2[x,y]$ with the following properties:

$I$ is generated by two homogeneous elements;
$I$ is invariant under the $SL_2(\mathbb{F}_2)$-action on $\mathbb{F}_2[x,y]$ (given  by extending the action on the two dimensional sub vector space spanned by $x,y$).;
The quotient $\mathbb{F}_2[x,y]/I$ is finite.

So far the only examples I know are $(x^n,y^n)$ and $(x^3,x^2+xy+y^2)$. The quotient $\mathbb{F}_2[x,y]/(x^3,x^2+xy+y^2)$ is the cohomology ring $H^*(S^3/Q_8;\mathbb{F}_2)$ for the standard action of the quaternion group on the three sphere. 
Are these the only examples. Is it possible to classify all such ideals ?
Edit: Of course $(x^n,y^n)$ is only invariant when $n$ is a power of two (since $x\mapsto x, y\mapsto x+y$ is also an automorphism). So we have even fewer examples.

REPLY [4 votes]: One can see that $I=(f_1,\dots, f_s)$ is $G$-invariant iff $\alpha\cdot f_i \in I$ for each $\alpha \in G$ and $i$. From this one can prove easily that if $I,J$ are $G$-invariant, then so is $IJ$ and $I+J$. Thus, for example, $(x,y)^n$ are invariant for all $n\geq 0$. One also have that $I=(x^{2^k}, y^{2^k})$ is invariant as pointed out by YCor. So you can generate many other examples. A complete classification seems difficult though. 
For the case of two-generated ideals, one could use the above remark and the fact that "Frobenius commutes with linear change of variables" to show:
1) If $I=(f,g)$ is invariant then $J= (f^{2^k}, g^{2^k})$ is invariant. 
2) If $I=(f,g)$ is invariant and $\deg(f)> \deg(g)$ then $J=(f^{2^k}, g^l)$ is invariant with $l\leq 2^k$. 
For instance the ideal $(x^6, x^2+xy+y^2)$ is invariant.