We prove that any finite abelian group $G$ contains a collection of not too many subsets with a special structure, so that for every subset $A$ of $G$ with a small doubling, there is a member $F$ of the collection that is fully contained in the sumset $A+A$ and is not much smaller than it. Using this result we obtain improved bounds for the problem of estimating the typical independence number of sparse random Cayley or Cayley-sum graphs, and for the problem of estimating the smallest size of a subset of $G$ which is not a sumset. We also obtain tight bounds for the typical maximum length of an arithmetic progression in the sumset of a sparse random subset of $G$.