Let $f(x_1,\ldots,x_n)$ be an unknown homogeneous degree-$d$ polynomial in $\mathbb{Q}[x_1,\ldots,x_n]$. Suppose that a permutation group $G$ acts on $(\mathbb{Q}^n)^*$ so that $f({\bf x}) = 0$ if and only if $f(g \cdot {\bf x}) = 0$ for all $g \in G$. [In my particular case, $n=9$, $d=8$, and $G =$ the symmetry group on the edges of a triangular prism $\mathbb{Z}_2 \times S_3$]
I would like to use the tools of invariant theory to find generators of a minimal submodule of $\mathbb{Q}[x_1,\ldots,x_n]_d$ guaranteed to contain $f({\bf x})$. However, the polynomials may not be quite invariant; if $g \in G$ has even order, then $f(g \cdot {\bf x})$ might be $-f({\bf x})$.
Is there literature addressing this variation on invariant theory (or a simple trick I'm missing)? Thanks in advance!