Not really an answer, but this is already difficult in the complex case, when $C$ is a central
subgroup. For example, if $C = Z(G),$ and we induce a faithful irreducible $C$-module to $G,$ the number of distinct irreducible constituents is bounded above by the number of conjugacy classes of $G/C,$ but I don't know many other general statements- the number can be as low as $1$ (when we have a character of so-called ``central type").
To understand the general case of inducing an indecomposable $RC$-module to $G$ for a general commutative ring $R$ when $C$ ne not be central, Mackey decompoosition is certainly a helpful tool ( the induced module restricts freely to any subgroup meeting $C$ only in the identity, and more generally, the restriction to a subgroup $H$ of the induced module can be reasonably described by knowing how the original indecomposable restricts to $C \cap H,$ though precise details may require care. When working over $\mathbb{Z}_{p}$, Green's indecomposability theorem may also be useful.