It is useful to reformulate the question in its natural differential topology setting, leaving unneeded geometric considerations aside. It is also natural to consider the analog of the problem in all dimensions.

So assume that we are given a closed, orientable, connected, smooth $n$-manifold $X$, and a closed, orientable, connected, smooth, codimension-$2$ submanifold $B \subset X$. We adopt the basic the notation used in the question. Let $G$ be a finite group. For every group epimorphism $$\varphi \colon \pi_1(X-B) \to G$$ there is a closed, orientable, connected, smooth $n$-manifold $Y$ and a Galois (or ``regular'') ramified covering map $$f \colon Y \to X,$$
with deck transformation group $G$ that is branched at most over $B$.

Since $B$ is smooth, setting $R =f^{-1}(B) \subset Y$ we see that the restriction $$f|_R \colon R \to B$$ is an unramified cover. The question seeks an explicit description of this covering map.

Among the issues that arise when trying to give such an explicit description are that $R$ need not be connected, that $f|_R:R \to B$ need not be a Galois covering, and that $B$ and $X-B$ cannot have the same base point.

The additional piece of data needed to clarify things is the normal bundle of the branch set and its boundary, a circle bundle over $B$. With this extra piece of information one can effectively answer the question.
We will from this point of view

- Characterize when $R$ is connected;
- Characterize when $f$ is
actually ramified;
- Characterize when $R \to B$ is Galois;
- Show that on each component of $R$ the restriction of the branched covering is in fact always a Galois covering, with an explicit Galois group.

Let $N$ denote a small tubular neighborhood of $B$ in $X$, which has the structure of a $2$-disk bundle over $B$. Let $D$ denote a 2-disk fiber, with boundary $C = D \cap \partial N$, a linking circle to $B$. Then $\partial N$ is a circle bundle over $B$, with typical fiber $C$.

This circle bundle is determined by its Euler class in $H^2(B;\mathbb{Z})$ and determines an exact sequence of homotopy groups (in which we suppress mention of the required base points)
$$
1 \to \pi_2(\partial N) \to \pi_2(B) \to \pi_1(C) \to \pi_1(\partial N) \to \pi_1(B)\to 1.
$$
The image of $\pi_1(C)$ in $\pi_1(\partial N)$ lies in the center because of our orientability assumption. The only case in the dimension range $n\leq 4$ that $\pi_2(B)\neq 1$ is when $n=4$ and $B=S^2$. In all other low-dimensional cases it reduces to a central extension of $\pi_1(B)$ by $\mathbb{Z}$.

In general the assertion that $R$ is connected is the same as requiring that $f^{-1}(\partial N)$ be connected. And that translates into the homomorphism
$$
\varphi j_*:\pi_1(\partial N) \to G
$$
being surjective, where $j:\partial N \to X-B$ is the inclusion.

The condition that actual ramification occurs, translates into the condition that the homomorphism
$$
\varphi i_*:\pi_1(C) \to G
$$
is nontrivial, where $i:C \to X-B$ is the inclusion.

In general the image of $\varphi j_*:\pi_1(\partial N)\to G$ gives the group of deck transformations on any one of the path components of the pre-image of the circle bundle $\partial N$ in $Y$.
It follows that for each component $R_k$ of the pre-image of the branch set, the projection $R_k\to B$ is a Galois covering with group of deck transformations isomorphic to
$$
\varphi j_*(\pi_1(\partial N))/ \varphi i_*(\pi_1(C)).
$$

The components of $R$ are permuted transitively by the action of $G$ on $Y$. The full ramification covering $R\to B$ is the quotient map for the action of $G$ restricted to $R$. The covering $R\to B$ will be Galois if and only if the image $\varphi i_*(\pi_1(C))$ is a normal subgroup of $G$, in which case the group of the covering is $G/ \varphi i_*(\pi_1(C))$.

Note, by the way, that since the image of $\pi_1(C)$ is central in $\pi_1(\partial N)$, it follows that if there is nontrivial ramification and $G$ has trivial center, then the pre-image of the branch set cannot be connected.

4more comments