Conjugacy growth of finitely generated groups

We show that every non-decreasing function $f\text{\hspace{0.17em}⁣}:\mathbb{N}\to \mathbb{N}$ 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.