Consider a function $F: \mathbb{F}_2^d \to \mathbb{Z}^n = (f_1,\ldots,f_n)$ with the property that if $y \in \mathbb{F}_2^d$ is a rotation of $x \in \mathbb{F}_2^d$, i.e. $y$ is $x$ permuted by an element of the cyclic group generated by $(1 \ 2 \ldots d)$, then F(x)=F(y), and the additional constraint that $f_i$ is of the form
$ f_i(x) = \sum\limits_{j=1}^d \alpha_jx_j $.
There are obvious examples of such functions, for instance $F(x) = \sum\limits_{1}^d x_i$. For any $x \in \mathbb{F}_2^d$ and $\pi \in S_d$, $F(x)=F(\pi x)$. We say that such a function is fixed under $S_d$. However, as you might expect from my initial explanation, I am interested in such a function which is fixed strictly under rotations, that is for any permutation $\pi$ which is not a rotation, then $F(x) \neq F(\pi x)$, or at least fixed under rotations and not fixed under all of $S_d$.
Do such functions exist? I am having difficulty finding any example which is not the one I gave above.