Let $\Gamma$ be a group and $\Gamma_1\supset\Gamma_2\supset\dots$ subgroups of finite index, such that $\bigcap_{j=1}^\infty \Gamma_j=\{1\}$. Let $1\ne\gamma\in\Gamma$ and let $[\gamma]=[\gamma]_\Gamma$ denote the conjugacy class of $\gamma$ in $\Gamma$. Is it true that there exists $n\in\mathbb N$ such that $[\gamma]\cap\Gamma_n=\emptyset$? This is the case, if every $\Gamma_n$ is normal in $\Gamma$, but what about the general case?

If it is not true, are there non-trivial estimates of the number of $\Gamma_n$ conjugacy classes within $[\gamma]$? More precisely, let $c_n$ denote the cardinatlity of the set $\big([\gamma]\cap\Gamma_n\big)/\Gamma_n$, where $\Gamma_n$ acts by conjugation. Does $\frac{c_n}{[\Gamma:\Gamma_n]}$ tend to zero?

If any of this does not hold in general, are there interesting classes of groups, for which it holds?