Conjugacy growth of finitely generated groups

We show that every non-decreasing function $f\colon \mathbb N\to \mathbb N$ bounded from above by $a^n$ for some $a\ge 1$ can be realized (up to a natural equivalence) as the conjugacy growth function of a finitely generated group. We also construct a finitely generated group $G$ and a subgroup $H\le G$ of index 2 such that $H$ has only 2 conjugacy classes while the conjugacy growth of $G$ is exponential. In particular, conjugacy growth is not a quasi-isometry invariant.
Comment: The published version of this paper contained an inaccuracy in the proof of Corollary 5.6, which was later corrected in Corrigendum to "Conjugacy growth of finitely generated groups", Adv. Math. 294 (2016), 857-859. This version incorporates all necessary corrections