The active control of internal transport barriers (ITBs) is an important issue to achieve high performance plasma in a fusion reactor. A critical challenge of ITB control is to increase the ITB position. The ITBs with internal kink modes (IKMs), such as fishbone (FB) instability and long-live mode (LLM) with mode number of m/n = 1/1 are frequently observed on HL-2A tokamak in neutral beam heated discharges. The correlation of FB instability/LLM with ITBs is analyzed in order to extend the ITB radius. It has been revealed that FB instability and LLM are often excited after the ITB formation. Therefore, FB instability and LLM play no role in triggering ITBs on HL-2A tokamak. On the other hand, they may slow down the outward radial expansion and then shrink the foot position of ITB, and damp the gradient growth of ion temperature and rotation velocity. Since the perturbation of LLM is weaker than that of FB instability, the shrinking effect of ITB foot and braking effect on gradient growth are slighter than those of FB instability. Compared with the LLM, FB instability routinely appears in plasmas with lower density, higher heating power and lower plasma current. In addition, large ITBs without IKMs are also discussed on HL-2A tokamak. The large ITB is the largest one, the FB ITB is the strongest one and the LLM ITB is the widest one in three ITBs, where the 'large', 'strong' and 'wide' qualifications correspond to ITB position ρITB, the normalized temperature gradient R/LT, and its width W/a. Therefore, the large ITB position may be obtained if the IKMs are effectively controlled in a tokamak.