Suppose I have a quotient $X \to S$ by a finite abelian group $G$ action (I have several cases, but in all of them the group $G$ and the action could be written explicitly), where $X,S$ are surfaces (smooth and path connected) over the complex numbers field. I also know $\pi_1(X)$, which is $\mathbb{Z}^4$ in my case and that the covering $X \to S$ is unramified. Is this data enough to compute $\pi_1(S)$? As far as I understand, it is known from general theory of coverings, that there is an injection $\pi_1(X) \to \pi_1(S)$, so I believe that $\pi_1(S)$ should be of the form $\pi_1(X) \rtimes G$, but I do not know what can I use to prove it (if this is correct).

Any input or advice is appreciated, thank you in advance.