Based on the types of isomerism in coordination compounds: (A) $[Co(NH_3)_5(NO_2)]Cl_2$ contains the ambidentate ligand $NO_2^-$, which can bind to the central metal atom through either nitrogen or oxygen. This leads to linkage isomerism . (A $\rightarrow$ II) (B) $[Co(NH_3)_5(SO_4)]Br$ can exchange its counter ion ($Br^-$) with a ligand ($SO_4^{2-}$) to form a different complex ion, $[Co(NH_3)_5Br]SO_4$. This is a characteristic of ionization isomerism . (B $\rightarrow$ III) (C) $[Co(NH_3)_6][Cr(CN)_6]$ consists of both complex cationic and complex anionic entities. The interchange of ligands between the two metal ions gives rise to coordination isomerism . (C $\rightarrow$ IV) (D) $[Co(H_2O)_6]Cl_3$ involves water as a solvent molecule that can act as a ligand directly bonded to the metal ion or exist as free solvent molecules in the crystal lattice, exhibiting solvate (hydrate) isomerism . (D $\rightarrow$ I)

Therefore, the correct match is (A)-(II), (B)-(III), (C)-(IV), (D)-(I).