ag.algebraic geometry – Necessary and sufficient condition for the lifting of a proper map to be proper

For $i=1,2$ let $(X_i,v_i)$ be two connected topological manifolds, two subgroups $H_ileq pi_1(X_i,v_i)$, two coverings $q_{H_i}:big(overline{X_i}(H_i),overline{v_i}big)to (X_i,v_i)$ corresponding to $H_i$.

Consider a proper map $f:(X_1,v_1)to (X_2,v_2)$ such that $f_#(H_1)leq H_2$ i.e. we have a lift $overline{f}:big(overline{X_1}(H_1),overline{v_1}big)to big(overline{X_2}(H_2),overline{v_2}big)$ with $q_{H_2}circoverline f=fcirc q_{H_1}$.

$require{AMScd}$
begin{CD}
left(overline{X_1}(H_1),overline{v_1}right) @>displaystyleoverline f>> left(overline{X_2}(H_2),overline{v_2}right)\
@Vq_{H_1} V V @VV q_{H_2}V\
left(X_1,v_1right) @>>displaystyle f> left(X_2,v_2right)\
end{CD}

Is there any necessary and sufficient condintions on $f, H_1,H_2$ such
that $overline f$ is a proper map?