Path-class construction of a subgroup covering

For a path-connected, locally path-connected, semilocally simply connected space $X$, base point $x_0$, and subgroup $H\le {\pi }_1(X,x_0)$, take paths starting at $x_0$ modulo the relation $\alpha \sim \beta$ when their endpoints agree and $[\alpha {\beta }^{-1}]\in H$. Projection sends a class to its endpoint. Over a path-connected neighbourhood whose loops die in $X$, classes $[\alpha \eta ]$, with $\eta$ inside that neighbourhood, form a sheet mapping homeomorphically to it. These sheets define a covering space. A loop lifts closed precisely when its class belongs to $H$, so the injective induced fundamental group map has image $H$.