  Let $(X, +)$ denote $(\mathbb{R}, +)$ or $(2^{\omega}, +_2)$. We prove that
for any meagre set $F \subseteq X$ there exists a subgroup $G \le X$ without
the Baire property, disjoint with some translation of F. We point out several
consequences of this fact and indicate why analogous result for the measure
cannot be established in ZFC. We extend proof techniques from the work of
Ros{\l}anowski and Shelah [1].
