Progression-free sets in class $2$ and arbitrary finite groups

We study progression-free sets in arbitrary finite groups. We transfer sharp upper bounds from abelian groups to nilpotent groups of class $2$, and from nilpotent groups to arbitrary groups. Combining a recent result of Raghavan with the author's theorem on large subgroups of class $2$ gives \[ r_3(G) \, \leq \, C \, |G|\exp\left( -c\frac{(\log|G|)^{1/6}}{(\log\log|G|)^{1/3}}\right) \] for every finite group as $|G|\to\infty$. This improves the estimates obtained by using the abelian subgroup supplied by Pyber's theorem.