Let $G: \mathbb{R}\rightarrow [0,1]$ be a cumulative distribution function (CDF) symmetric about zero, i.e., $G(x)=1-G(-x)$ at each $x\in \mathbb{R}$.
Take some real numbers $\mu_1,\mu_2$.
Consider the two CDFs
(1) $F_1:x\mapsto G(x-\mu_1)$
(2) $F_2: x\mapsto G(x-\mu_2)$
The functions (1) and (2) are still symmetric but not about zero (because of the shifting by $\mu_1,\mu_2$)
Question: suppose $\mu_1\neq \mu_2$. Can we say something about how the moments of $F_1,F_2$ will differ? In other words, is there a deterministic relation (depending on $\mu_1,\mu_2$) between the moments of $F_1$ and the moments of $F_2$?