Non-meagre subgroups of reals disjoint with meagre sets

Let (X,+) denote (R,+) or (2ω,+2). We prove that for any meagre set F⊆X there exists a subgroup G≤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 [1].