  Let $\mathscr J$ be the set of inner functions whose derivatives lie in
Nevanlinna class. In this note, we show that the natural map $F \to
\text{Inn}(F'): \mathscr J/\text{Aut}(\mathbb{D}) \to \text{Inn}/S^1$ is is
injective but not surjective. More precisely, we show that that the image
consists of all inner functions of the form $BS_\mu$ where $B$ is a Blaschke
product and $S_\mu$ is the singular factor associated to a measure $\mu$ whose
support is contained in a countable union of Beurling-Carleson sets. Our proof
is based on extending the work of D. Kraus and O. Roth on maximal Blaschke
products to allow for singular factors. This answers a question raised by K.
Dyakonov.
