Induced birational transformations on O'Grady's sixfolds

We introduce the notion of induced birational transformations of irreducible holomorphic symplectic sixfolds of the sporadic deformation type discovered by O'Grady. We give a criterion to determine when a manifold of $OG_6$ type is birational to a moduli space of sheaves on an abelian surface. Then we determine when a birational transformation of the moduli space is induced by an automorphism of the abelian surface. Referring to the Mongardi--Rapagnetta--Sacc\'{a} birational model of manifolds of $OG_6$ type, we give a result to determine when a birational transformation is induced at the quotient. We give an application of these criteria in the nonsymplectic case.