Let $n$ be a nonnegative integer. For each composition $α$ of $n$, Berg $\textit{et al.}$ introduced a cyclic indecomposable $H_n(0)$-module $\mathcal{V}_α$ with a dual immaculate quasisymmetric function as the image of the quasisymmetric characteristic. In this paper, we study $\mathcal{V}_α$'s from the homological viewpoint. To be precise, we construct a minimal projective presentation of $\mathcal{V}_α$ and a minimal injective presentation of $\mathcal{V}_α$ as well. Using them, we compute ${\rm Ext}^1_{H_n(0)}(\mathcal{V}_α, {\bf F}_β)$ and ${\rm Ext}^1_{H_n(0)}( {\bf F}_β, \mathcal{V}_α)$, where ${\bf F}_β$ is the simple $H_n(0)$-module attached to a composition $β$ of $n$. We also compute ${\rm Ext}_{H_n(0)}^i(\mathcal{V}_α,\mathcal{V}_β)$ when $i=0,1$ and $β\le_l α$, where $\le_l$ represents the lexicographic order on compositions.