Borel sets without perfectly many overlapping translations, III

We expand the results of Roslanowski and Shelah arXive:1806.06283 , arXive:1909.00937 to all perfect Abelian Polish groups $(H,+)$. In particular, we show that if $\alpha<\omega_1$ and $4\leq k<\omega$, then there is a ccc forcing notion adding a $\Sigma^0_2$ set $B\subseteq H$ which has $\aleph_\alpha$ many pairwise $k$--overlapping translations but not a perfect set of such translations.