The notion of naive liftability of DG modules is introduced in [9] and [10]. In this paper, we study the obstruction to naive liftability along extensions A→B of DG algebras, where B is projective as an underlying graded A-module. We show that the obstruction to naive liftability of a semifree DG B-module N is a certain cohomology class in Ext1B(N,N⊗BJ), where J is the diagonal ideal. Our results on obstruction class enable us to give concrete examples of DG modules that do and do not satisfy the naive lifting property.
