Rank-two reflexive sheaves on the projective space with second Chern class equal to four

We study rank-two reflexive sheaves on $\mathbb{P}^3$ with $c_2 =4$, expanding on previous results for $c_2\le3$. We show that every spectrum not previously ruled out is realized. Moreover, moduli spaces are studied and described in detail for $c_1=-1$ or $0$ and $c_3\ge8$.