Serway's 10ed says that Newton's second law of rotational ($\sum \tau_{z} = I_{z}\alpha_z $) is true when there is combined translation and rotation as long as the moving axis (1) passes through the center of mass and (2) is an axis of symmetry. Sears-Zemanski (edition 12) also adds that (3) the axis must not change direction.
I don't know how to prove it, but I do not find any of these conditions strictly necessary. In this post it has been mentioned that at a given moment, the movement of a rigid body can be described by a translation of any chosen point, plus a rotation about that point, so the axis of rotation is an arbitrary matter. Condition 3 doesn't seem necessary to me either ... only that a time-varying direction axis would make the counts much more complex, because the equations would change at every instant.
My hypothesis is that these conditions are more a description of the type of exercises the books are limited to rather than a necessity for applying the equations. I am right?