The projective cover of tableau-cyclic indecomposable $H_n(0)$-modules

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.