Consider the additive group $G=\sum_{i\in I}\mathbb{Z}_2$, and for every $A\subseteq G$ put $A-A=\{a_1-a_2: a_1,a_2\in A\}$, $$ S_A:=\{B\subseteq G: (A-A)\cap(B-B)=\{0\} , (A-A)+B=G\} $$ Now, is it true that:
(a) For every $A\subseteq G$ all elements of $S_A$ have the same cardinal number?
Note: It seems that answer of (a) is positive if $I$ is finite (by using the GAP) but not sure. Now if answer of (a) is negative, for some index set $I$, then what about the following statements?
(b) If $I$ is infinite then there exists a subset $A$ of $G$ such that $S_A$ contains both finite and infinite elements ;
(c) If $I$ is infinite then there exists a subset $A$ of $G$ such that $S_A$ contains only finite elements (i.e., $|B|$'s are finite) and $\{|B|: B\in S_A\}$ is unbounded above.