We study the $H_n(0)$-module $\mathbf{S}^σ_α$ due to Tewari and van Willigenburg, which was constructed using new combinatorial objects called standard permuted composition tableaux and decomposed into cyclic submodules. First, we show that every direct summand appearing in their decomposition is indecomposable and characterize when $\mathbf{S}^σ_α$ is indecomposable. Second, we find characteristic relations among $\mathbf{S}^σ_α$'s and expand the image of $\mathbf{S}^σ_α$ under the quasi characteristic in terms of quasisymmetric Schur functions. Finally, we show that the canonical submodule of $\mathbf{S}^σ_α$ appears as a homomorphic image of a projective indecomposable module.