We study the magnetic field controlled optical transmission observed in [Phys. Rev. Lett. 118, 263601 (2017)] with ultra-cold 88 Sr atoms trapped in a one-dimensional optical lattice inside an optical cavity. We show that this phenomenon can be understood with a Jaynes-Cumming-like model for the atoms with two Zeeman-split excited states. This model yields three peaks in the transmission spectrum associated with three singly excited dressed states. The transmission is controlled by adjusting the photonic components in the dressed states via the Zeeman splitting of the atomic levels. If the atomic ensemble is incoherently pumped, lasing can be achieved in a superradiant crossover regime. Without the magnetic field the lasing relies on bright dressed states featuring a spectrum with line-width down to 5 kHz. However, in the presence of the magnetic field, the lasing relies on the dressed states, which are originally uncoupled but acquire photonic component due to the Zeeman splitting, and can yield a spectrum with line-width down to 2 Hz.