Apologies in advance for what I am asking might be too trivial, I am not a mathematician.
I have a function $V(x_1,\dots,x_n)$ that could have, at most, a hyperoctahedral (signed permutations) symmetry given by $H = S_2^n \rtimes S_n$. For those who do not know, acting with a group element on a vector space element permutes the vector element indices and changes the signs of some elements. I say that it can have at most this symmetry because $V$ might have fewer symmetries, i.e. be just a subgroup of the hyperoctahedral group.
My goal is to find this subgroup and to check whether two vector elements are the same under the subgroup.
To solve the first problem I generated all group elements $g$ and checked which group elements keep $V$ the same, i.e. $V(x_i) = V(g \cdot x_i)$. Is this the optimal way? Do I have to check all $2^n n!$ ($|H| = 2^n n!$) elements or would checking only the group generators be enough?
For the second part of my question I do not have yet have a clear answer. This is my attempt. Inspired by the elementary symmetric polynomials (example 1.4 of https://www1.mat.uniroma1.it/people/rogora/pdf/Enrico.pdf). I defined a polynomial $$ \phi(t):=\left(t-x_1^2\right)\left(t-x_2^2\right) \cdots\left(t-x_n^2\right)=t^n-\sigma_1 t^{n-1}+\sigma_2 t^{n-2}-\cdots+(-1)^n \sigma_n $$ where the $\sigma_i$ are given by $$ \begin{aligned} \sigma_1 & =x_1^2+x_2^2+\cdots+x_n^2 \\ \sigma_2 & =x_1^2 x_2^2+x_1^2 x_3^2+\cdots+x_1^2 x_n^2+x_2^2 x_3^2+\cdots+x_{n-1}^2 x_n^2 \\ \vdots & \vdots \\ \sigma_r & =\sum_{i_1<i_2<\cdots i_r} x_{i_1}^2 x_{i_2}^2 \cdots x_{i_r}^2 \\ \vdots & \vdots \\ \sigma_n & =x_1^2 x_2^2 \cdots x_n^2 \end{aligned} $$
It is obvious that this polynomial is invariant under H. Could I use $\sigma_i$ as invariants of $H$? If two elements $x,y\in V$ have the same $\sigma_i$, can I conclude that they are the same under the group symmetries, i.e. there exists a $h\in H$ such that $x = h\cdot y$?