  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.
