  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
$A\to B$ 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 Ext$^1_B(N,N\otimes_B J)$,
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.
