The classical Neumann lemma states that if a group is covered by finitely many cosets, then at least one of these cosets is the coset of a subgroup of finite index. (Actually, the lemma says more, namely that the group is covered by the cosets of subgroups of finite index.)

I wonder if there is an infinitary version of the lemma in the following sense: Suppose that $G$ is a group and $G=\bigcup_{i<\alpha}x_iH_i$, where the $H_i$ are subgroups and $\alpha$ is an ordinal less than some big cardinal $\kappa$. (Maybe even $\kappa$ is strongly inaccessible.) Is it necessarily the case that there is some $i$ for which the index of $H_i$ in $G$ is less than $\kappa$?