Say we have $X$ a manifold with a compact exhaustion of embedded submanifolds $X=\cup K_n$ with $K_n\subset K_{n+1}$. Let $H\subset \pi_1(X)$ a infinite index subgroup that is finitely generated, therefore we know that $H$ will be represented inside some $K_N$ for $N$ big enough.
Can we then compute the corresponding cover $X_H$ by doing for the exhaustion?
For example say that the covers for $n\geq N$ $(K_n)_H$ are all homeomorphic can we then say that $X_H$ is also homeomorphic to this?