Let \mathcal{Y}\xrightarrow {{\psi}} \mathcal{X} \xrightarrow {\varphi} \mathbb{P}^{1} be a sequence of covers of compact Riemann surfaces. In this work we study and completely characterize the Galois group \mathfrak{G}(\varphi\circ\psi) of \varphi\circ\psi under the following properties: \varphi is a simple cover of degree m and \psi is a Galois unramified cover of degree n with abelian Galois group of type (n_1,n_2,\dots,n_s) . We prove that \mathfrak{G}(\varphi\circ\psi) \cong ({\mathbb Z}_{n_1} \times {\mathbb Z}_{n_2} \times \cdots \times {\mathbb Z}_{n_s})^{m-1} \rtimes {\bf S}_m . Furthermore, we give a natural geometric generator system of \mathfrak{G}(\varphi\circ\psi) obtained by studying the action on the compact Riemann surface \mathcal{Z} associated to the Galois closure of \varphi\circ\psi.