The locally trivial monodromy group is an important deformation invariant for irreducible symplectic varieties and plays a fundamental role in the study of their birational geometry. In this talk, I will give a description of the locally trivial monodromy group of moduli spaces of semistable sheaves on Abelian surfaces, with non-primitive Mukai vector. The outcome is that the locally trivial monodromy group of a singular moduli space of this type is isomorphic to the monodromy group of a smooth moduli space, extending Markman’s and Mongardi’s description to the non-primitive case. As a geometric application, I will sketch a proof for the SYZ conjecture for any singular moduli space of this type.