In this paper we study ground-states of the fractional Gierer–Meinhardt system on the real line, namely the solutions of the problem $$\begin{aligned} \left\{ \begin{array}{lll} (-\Delta )^su+u-\frac{u^2}{v}=0~&{}\quad \mathrm {in}~\mathbb {R},\\ (-\Delta )^sv+\varepsilon ^{2s}v-u^2=0~&{}\quad \mathrm {in}~\mathbb {R},\\ u,v>0,\quad u,v\rightarrow 0,&{}\quad \mathrm {as}~|x|\rightarrow \infty . \end{array}\right. \end{aligned}$$ We prove that given any positive integer k, there exists a solution to this problem for $$s \in [\frac{1}{2}, 1)$$ exhibiting exactly k bumps in its $$u-$$ component, separated from each other at a distance $$ O(\varepsilon ^{\frac{1-2s}{4s}})$$ for $$ s \in (\frac{1}{2}, 1)$$ and $$O(|\log \varepsilon |^{\frac{1}{2}})$$ for $$ s=\frac{1}{2}$$ , whenever $$\varepsilon $$ is sufficiently small. After suitable scaling, each bump of u is exactly the same as the unique solution of $$\begin{aligned} (-\Delta )^s U+U-U^2=0~\mathrm {in}~\mathbb {R},\quad 0<U(y)\rightarrow 0~\mathrm {as}~|y|\rightarrow \infty . \end{aligned}$$