Quasi-projective and quasi-liftable characters

We study ordinary characters of a finite group G which vanish on the p-singular elements for a fixed prime p dividing the order of G. Such characters are called quasi-projective. We show that all quasi-projective characters of G are characters of projective modules if and only if the ordinary irreducible characters of G can be ordered in such a way that the top square fragment of the decomposition matrix is diagonal. Finally, we prove that the number of indecomposable quasi-projective characters of G is finite and characterize them in case of blocks with cyclic defect groups.