Let $a$ be a positive integer which is not a perfect $b$ -th power with $b\geq 2$ , $q$ be a prime number and $Q_a(x;q^i,j)$ be the set of primes $p\leq x$ such that the residual order of $a (\mathrm{mod} p)$ in $(\mathbf{Z}/p\mathbf{Z})^\times$ is congruent to $j$ modulo $q^i$ . In this paper, which is a sequel of our previous papers [1] and [6], under the assumption of the Generalized Riemann Hypothesis, we determine the natural densities of $Q_a(x;q^i,j)$ for $i\geq 3$ if $q=2$ , $i\geq 1$ if $q$ is an odd prime, and for an arbitrary nonzero integer $j$ (the main results of this paper are announced without proof in [3], [7] and [2]).