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