Computing critical points for algebraic systems defined by hyperoctahedral invariant polynomials

Let $\mathbb{K}$ be a field of characteristic zero and $\mathbb{K}[x_1, \dots, x_n]$ the corresponding multivariate polynomial ring. Given a sequence of $s$ polynomials $\mathbf{f} = (f_1, \dots, f_s)$ and a polynomial $\phi$, all in $\mathbb{K}[x_1, \dots, x_n]$ with $s