Let $F$ be the set of bijective Borel-measurable functions $f \colon [0,1] \to [0,1]$ that preserve the Lebesgue measure.

Is it the case that for every non-Lebesgue-measurable set $A \subset [0,1]$, there exists a countable family $\{f_n\}_{n \in \mathbb{N}} \subset F$ such that $\ \bigcup_{n \in \mathbb{N}} f_n(A)\,$ contains a Lebesgue-measurable set of positive measure?

If so, then there is no natural way to extend the Lebesgue measure to include more null sets.

(Conversely, if not, then it seems reasonable to regard all the counterexemplary sets as "kind-of-null sets".)