Coverings for $4$-dimensional almost complex manifolds with non-degenerate torsion

An almost complex manifolds $(M^4,J)$ of real dimension 4 with non-degenerate torsion bundle admit a double absolute parallelism and it is provided the classification of homogeneous $(M^4,J)$ having an associated non-solvable Lie algebra. We extend such a classification to the analysis of the manifolds having an associated solvable Lie algebra, up-to-coverings. Moreover, for homogeneous $(M^4,J)$ we provide examples with connected and non-connected double covering, thus proving that in general the double absolute parallelism is not the restriction of two absolute parallelisms. Furthermore, it is given the definition of a natural metric induced by the absolute parallelisms on $(M^4,J)$ and an example of an almost complex manifold with non-degenerate torsion endowed with that metric such that it becomes an almost K\"ahler manifold.