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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.