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Hybrid jump linear systems (HJLS's) were introduced to study the properties of supervisory control systems. In a recent publication, a mean square (MS) stability test was introduced for HJLS's with Moore output maps and i.i.d. inputs. In this paper, a similar test is developed for HJLS's of this type but with Markov inputs. The test requires one to first construct an auxiliary HJLS with equivalent stability properties. Then, a particular matrix, A, is constructed taking into consideration the auxiliary system's structure and associated Markov kernel. Finally, the spectral radius of A is compared against 1 to determine the stability of the original HJLS.