There has been a growing trend to use the homology of simplicial complexes to study complex data structures because of its resilience to deformation and noise. When the data have underlying symmetry, the corresponding simplicial complex is equipped with a group action that can be leveraged to improve the efficiency of computation. Carbone, Nanda, and Naqvi described a geometric compression algorithm that compresses an equivariant simplicial complex to a complex of groups, a framework that encodes the local structure of a group action. Given a simplicial complex $X$ equipped with a regular action of a finite group $G$. In this dissertation, the question of recovering homology with coefficients in a field $\mathbb{F}$ of $X$, using its quotient space $X/G$ and a complex of groups, is explored. An alternative formulation based on group rings of the complexes of groups is developed. A complex of groups is then combined with the standard simplicial boundary matrices of $X/G$ to construct the $G$-boundary matrices, which are matrices over $\mathbb{F}G$. In the case $G = \mathbb{Z}_k,$ the group ring $\mathbb{F}G$ is commutative and matrices over $\mathbb{F}G$ admits a Smith normal form. The relation between matrix operations on a $G$-boundary matrix and that on a corresponding boundary matrix of $X$ is then investigated. Through these steps, a new algebraic approach has been developed to compute the homology of a simplicial $G$-complex $X$.