Every subgroup acts on the universal cover with a connected quotient covering that realizes it

Fix a universal cover $p:(\tilde{B},{\tilde{b}}_0)\to (B,b_0)$ by [F1], and use [F2] to regard $H$ as a subgroup of its deck group. Over an evenly covered neighbourhood of $b\in B$, choose the sheet containing a given $\tilde{b}$. A nonidentity deck transformation sends it to a different sheet: otherwise it would send the unique point over $b$ in that sheet to itself and hence be the identity by [F7]. Thus [F5] holds, so the restricted $H$-action is a covering-space action.

Let $q:\tilde{B}\to E_H:=\tilde{B}\mathrm{/}H$ be the orbit map, which is a covering by [F3]. Since $p$ is constant on $H$-orbits, it induces $p_H:E_H\to B$. Over an evenly covered $W\subseteq B$, the $H$-orbits of the sheets of $p^{-1}(W)$ have disjoint images under $q$, and on each such image $p_H$ is identified with the homeomorphism from any representative sheet to $W$. Hence $p_H$ is a covering by [F6]. The path-connected space $\tilde{B}$ maps continuously and surjectively to $E_H$, so $E_H$ is path-connected, with basepoint $e_H=q({\tilde{b}}_0)$.

For a loop $\alpha$ at $b_0$, its lift to $E_H$ from $e_H$ is $q\circ \tilde{\alpha }$, where $\tilde{\alpha }$ is its universal lift. This lift closes exactly when the universal endpoint lies in the $H$-orbit of ${\tilde{b}}_0$, which by [F2] and [F4] holds exactly when $[\alpha ]\in H$. A loop class is in $(p_H{)}_{\ast }{\pi }_1(E_H,e_H)$ exactly when it has a closed lift to $E_H$, so the induced subgroup is precisely $H$.