Rearrangement of conditionally convergent series on a small set

We consider ideals II of subsets of the set of natural numbers such that for every conditionally convergent series ∑n∈ωan∑n∈ωan and every r∈R¯ there is a permutation πr:ω→ω such that ∑n∈ωaπr(n)=r∑n∈ωaπr(n)=r and{n∈ω:πr(n)≠n}∈I. We characterize such ideals in terms of extendability to a summable ideal (this answers a question of Wilczyński). Additionally, we consider Sierpiński-like theorems, where one can rearrange only indices with positive anan.