Nagao-Nakajima introduced counting invariants of stable perverse coherent systems on small resolutions of Calabi–Yau threefolds and determined them on the resolved conifold. Their invariants recover DT/PT invariants and Szendröi’s non-commutative invariants in some chambers of stability conditions. In this paper, we study an analogue of their work on Calabi–Yau fourfolds. We define counting invariants for stable perverse coherent systems using primary insertions and compute them in all chambers of stability conditions. We also study counting invariants of local resolved conifold defined using torus localization and tautological insertions. We conjecture a wall-crossing formula for them, which upon dimensional reduction recovers Nagao-Nakajima’s wall-crossing formula on resolved conifold.