I understand the correspondence between the subgroups of a fundamental group $\pi_1(X)$ and the covering spaces of $X$.
However, I do not understand what is implied about the fundamental groups of those covering spaces. Are they the associated subgroups of $X$?
In other words, if subgroup $A\subset \pi_1(X)$ is associated with a covering space $Y$, is it true that $\pi_1(Y)=A$? (Or is it $\pi(X)/A$? Or is it not determined by $\pi(X)$ at all?)