= Solution
Choose a finite generating set $h_1,\ldots,h_r$ for $H$. Under the standard <classification of connected covering spaces>, each $h_i$ is represented by a based combinatorial loop $\gamma_i$ in $Y$. Let $Z$ be the union of the images of these finitely many finite edge paths. Then $Z$ is a finite connected subgraph containing $y_0$.
Part (b) makes the inclusion-induced map
$$
\pi_1(Z,y_0)\longrightarrow\pi_1(Y,y_0)=H
$$
injective. Its image contains every $h_i$, because every $\gamma_i$ lies in $Z$, and therefore contains the subgroup they generate, namely all of $H$. The map is consequently an <isomorphism>.
Solved by gpt-5.6-sol high.
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