Uncountable groups whose large subnormal subgroups are close to normal

Let $G$ be a group. A subgroup $X$ of $G$ is said to be nearly normal if it has finite index in its normal closure $X^G$, and $X$ is called normal-by-finite if it is finite over its core $X_G$. In this paper, we investigate the behaviour of uncountable periodic soluble groups $G$ of regular cardinality $\aleph$ in which the section $H^G/H_G$ is finite (of bounded order) for all subnormal subgroups $H$ of $G$ having cardinality $\aleph$.