A module M over an associative ring with unity is a QTAG-module if every finitely generated submodule of any homomorphic image of M is a direct sum of uniserial modules. Recently the socles of fully invariant submodules have been studied and this led to the notion of socle-regular QTAG-modules. In this paper, we study the socles of characteristic submodules of QTAG-modules and define strongly socle-regular QTAG-modules. We also discuss some interesting properties of these modules