On the distribution of the order over residue classes

For a fixed rational number $g\not\in \{-1,0,1\}$ and integers $a$ and $d$ we consider the set $N_g(a,d)$ of primes $p$ such that the order of $g$ modulo $p$ is congruent to $a\,({\rm mod~}d)$. Under the Generalized Riemann Hypothesis (GRH), it can be shown that the set $N_g(a,d)$ has a natural density $\delta_g(a,d)$. Arithmetical properties of $\delta_g(a,d)$ are described, and $\delta_g(a,d)$ is compared with $\delta(a,d)$: the average density of elements in a field of prime characteristic having order congruent to $a\,({\rm mod~}d)$. It transpires that $\delta_g(a,d)$ has a strong tendency to be equal to $\delta(a,d)$, or at least to be close to it.