  Matter fields don't necessarily have to share the symmetries with the
spacetime they live in. When this happens, we speak of the symmetry inheritance
of fields. In this paper we classify the obstructions of symmetry inheritance
by the scalar fields, both real and complex, and look more closely at the
special cases of stationary and axially symmetric spacetimes. Since the
symmetry noninheritance is present in the scalar fields of boson stars and may
enable the existence of the black hole scalar hair, our results narrow the
possible classes of such solutions. Finally, we define and analyse the symmetry
noninheritance contributions to Komar mass and angular momentum of the black
hole scalar hair.
