Maps preserving the sum-to-difference ratio in characteristic $p$

Given a field $\mathbb{F}$, we introduce a novel group $SD(\mathbb{F})$ of its self-maps: the solutions $f\text{\hspace{0.17em}⁣}:\mathbb{F}\twoheadrightarrow \mathbb{F}$ to the functional equation $f((x+y)\mathrm{/}(x-y))=(f(x)+f(y))\mathrm{/}(f(x)-f(y))$ for all $x\mathrm{\ne }y$ in $\mathbb{F}$. We compute this group for all fields algebraic over ${\mathbb{F}}_p$. In particular, this group distinguishes ${\mathbb{F}}_5$ among all finite fields ${\mathbb{F}}_q$, and in fact among all subfields of $\overline{{\mathbb{F}}_q}$.