We consider the classes $T_n$ of permutations of degree $n$ whose cycle lengths belong to a set $A\subseteq\mathbb N$, where the set $A$ is completely determined by a given regularly varying function $g(t)$ and a finite union $\Delta$ of intervals from $[0,1]$. We find the asymptotics of the number of elements of $T_n$ as $n \to\infty$. The limit theorems on the total number of cycles and the number of cycles of a fixed length in random permutations uniformly distributed on $T_n$ are proved. This paper continues the investigations we started in [ibid. 1, No. 1, 105–116 (1991; Zbl 0728.05004)].