Suppose we have an infinite compact (Hausdorff) group $G$, and a subgroup $H\leq G$ which is meagre.

Can $H$ always be covered by a countable family of nowhere dense sets $H_n$ such that $H_n^2$ is still nowhere dense, for each $n$?

Clearly, we can assume that $H$ is dense in $G$. I believe it holds when $H$ is (cointained in) a meagre $F_\sigma$ subgroup, because then we can just take for $H_n$ the closed nowhere dense sets which add up to $H$, and then use the fact that a meager closed set is nowhere dense.

On the other hand, just taking for $H_n$ a family of closed nowhere dense sets covering $H$ is not good enough -- for instance if any $H_n=H_n^{-1}$ is not null, its square will have nonempty interior by Steinhaus theorem.

I don't have much experience with abstract compact groups, so I have hard time even imaging $H,G$ which do not satisfy the assumptions of the previous paragraph...