A generalization of groups with many almost normal subgroups

A subgroup $H$ of a group $G$ is called almost normal in $G$ if it has finitely many conjugates in $G$. A classic result of B. H. Neumann informs us that $|G:\mathbf{Z}(G)|$ is finite if and only if each $H$ is almost normal in $G$. Starting from this result, we investigate the structure of a group in which each non-finitely generated subgroup satisfies a property, which is weaker to be almost normal.