Let $G$ be a group, and let $H$ be a normal finite subgroup of $G$.
It is known that subgroups of the quotient group $G/H$ correspond to subgroups of $G$ containing $H$.
Does the statement still hold replacing "subgroups" by "finite subgroups"?
Clearly, every finite subgroup $F$ in $G$ is sent to a finite subgroup of $G/H$ via the projection $\pi: G \to G/H$. Does every finite subgroup of $G/H$ arise this way?
Let $K/H$ be a finite subgroup of $G/H$ (with $K$ a subgroup of $G$ containing $H$). Then the (finite) order of $K/H$ is the index $|K:H|$.
Then $K$ is finite, otherwise $|K:H|$ must be infinite since $H$ is finite, yielding a contradiction.