I have a question about surgery.

Let $G= \mathbb{Z}_m \times \mathbb{Z}$ and $M$ be a oriented 3-manifold with G-action. i.e. There exists a map $f\colon M/G \to BG$, where $BG$ is classifying space.($BG=K(G,1)=K(\mathbb{Z}_m \times \mathbb{Z},1)).$ Therefore, we can regard $(M/G,f)\in \Omega^{SO}_3(K(G,1))$, where $\Omega^{SO}_3(X)$ is (3-dimensional oriented) bordism group of $X$. By using Leray-Serre spectral sequence, we can easily see that $\Omega^{SO}_3(K(G,1))=H_3(G)=\mathbb{Z}_m$ (By using Kunneth fomula) Therefore, $\exists$ r>0 such that $r(M/G,f)$ is nullbordant over $G$. (we can choose $r =m$) That is, there exists a 4-manifold $V$ with G-action such that $\partial V = M$ and $\partial (V/G) = M/G$ and induced map $g\colon V/G\to BG$ is an extension of $f\colon M/G \to BG$.

$\underline{Question}$ : Can I do a surgery on $V/G$ to make $\pi_1(V/G)=G$ without losing $G$ action on $V$?