  Let $\alpha$ be a composition of $n$ and $\sigma$ a permutation in
$\mathfrak{S}_{\ell(\alpha)}$. This paper concerns the projective covers of
$H_n(0)$-modules $\mathcal{V}_\alpha$, $X_\alpha$ and
$\mathbf{S}^\sigma_{\alpha}$, which categorify the dual immaculate
quasisymmetric function, the extended Schur function, and the quasisymmetric
Schur function when $\sigma$ is the identity, respectively. First, we show that
the projective cover of $\mathcal{V}_\alpha$ is the projective indecomposable
module $\mathbf{P}_\alpha$ due to Norton, and $X_\alpha$ and the $\phi$-twist
of the canonical submodule $\mathbf{S}^{\sigma}_{\beta,C}$ of
$\mathbf{S}^\sigma_{\beta}$ for $(\beta,\sigma)$'s satisfying suitable
conditions appear as $H_n(0)$-homomorphic images of $\mathcal{V}_\alpha$.
Second, we introduce a combinatorial model for the $\phi$-twist of
$\mathbf{S}^\sigma_{\alpha}$ and derive a series of surjections starting from
$\mathbf{P}_\alpha$ to the $\phi$-twist of
$\mathbf{S}^{\mathrm{id}}_{\alpha,C}$. Finally, we construct the projective
cover of every indecomposable direct summand $\mathbf{S}^\sigma_{\alpha, E}$ of
$\mathbf{S}^\sigma_{\alpha}$. As a byproduct, we give a characterization of
triples $(\sigma, \alpha, E)$ such that the projective cover of
$\mathbf{S}^\sigma_{\alpha, E}$ is indecomposable.
