When can you "extend" subgroups? Suppose $H\le G\le G'$ are subgroups, with $H$ finite index in $G$. Can you always find a subgroup $H'\le G'$ of finite index such that $H'\cap G = H$?
Of course it's trivial if $G$ is finite index in $G'$, so we may suppose $[G':G] = \infty$.

For a counterexample, let $G'$ be an infinite simple group containing a nontrivial torsion element $g$, take $H=1$, $G=\langle g \rangle$. Since $G'$ is simple, it has no proper subgroups of finite index.
For example, you could take $G'$ to be the alternating group on a countable set or, for a more exotic example, a Tarski Monster.
