Partitions and sums with inverses in Abelian groups

Let G be an Abelian group and let A = { x ∈ G : 2 x ≠ 0 } be infinite. We construct a partition { A m : m < ω } of A such that whenever ( x n ) n < ω is a one-to-one sequence in A, g ∈ G and m < ω , one has ( g + FSI ( ( x n ) n < ω ) ) ∩ A m ≠ ∅ , where FSI ( ( x n ) n < ω ) = { ∑ n ∈ F ε n F x n : F ∈ P f ( ω )  and  ε n F ∈ { 1 , − 1 }  for all  n ∈ F } and P f ( ω ) is the set of finite nonempty subsets of ω.