We have classified in [7] all linear control systems and studied their controllability and control sets on the homogeneous spaces L\H of the 3-dimensional Heisenberg group H through its closed subgroups L. In this paper, we characterize the controllability and control sets of the induced linear control systems in the homogeneous spaces left. In particular, we focus on the singularity of the induced drift vector fields. Quite a technical analysis with many cases reveals the control sets. We give some geometric illustrations. (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.