Let $M$ be a Seifert fibered space over an orbifold $B$. Assume that $B$ is good and has infinite orbifold fundamental group. Then it is well known that there is a short exact sequence. $$1\to {\Bbb Z}\to \pi_1(M,m)\to \pi_1^{orb}(B,b)\to 1.$$ Where $\Bbb Z$ is generated by a fiber over a regular point $b\in B$. I would like to see this exact sequence induced in a different way.

Let $q:M\to B$ be the Seifert fibration and $p:\tilde B\to B$ be a good orbifold covering. Let $\tilde q:\tilde M\to \tilde B$ be the pull back of $q$ by $p$. Let $G$ and $\tilde G$ be the group of covering transformations of $p$ and $\tilde p:\tilde M\to M$ respectively. Now consider the translation Lie groupoids ${\cal G}(\tilde M, \tilde G)$ and ${\cal G}(\tilde B, G)$. This data then give a homomorphism $Q: {\cal G}(\tilde M, \tilde G)\to {\cal G}(\tilde B, G)$, which in turn induces a map on their classifying spaces $BQ: B{\cal G}(\tilde M, \tilde G)\to B{\cal G}(\tilde B, G)$. Now if we apply the fundamental group functor on the map $BQ$, for a regular point $b\in B$, then I believe that it induces the above exact sequence.

Am I correct in the above deduction?

My confusion is arrising from the understanding that since $\tilde q$ is a genuine fibration, then $BQ$ is a quasifibration. But it seems that is not the case, since the above exact sequence exists only for regular points in $B$ and not for all points. Here note that $M$ and $B$ both are aspherical and hence both $B{\cal G}(\tilde M, \tilde G)$ and $B{\cal G}(\tilde B, G)$ are aspherical.