A group version of stable regularity

We prove that, given $ϵ>0$ and $k\ge 1$, there is an integer $n$ such that the following holds. Suppose $G$ is a finite group and $A\subseteq G$ is $k$-stable. Then there is a normal subgroup $H\le G$ of index at most $n$, and a set $Y\subseteq G$, which is a union of cosets of $H$, such that $\mathrm{\mid }A△Y\mathrm{\mid }\le ϵ\mathrm{\mid }H\mathrm{\mid }$. It follows that, for any coset $C$ of $H$, either $\mathrm{\mid }C\cap A\mathrm{\mid }\le ϵ\mathrm{\mid }H\mathrm{\mid }$ or $\mathrm{\mid }C\setminus A\mathrm{\mid }\le ϵ\mathrm{\mid }H\mathrm{\mid }$. This qualitatively generalizes recent work of Terry and Wolf on vector spaces over ${\mathbb{F}}_p$.