We are concerned with the following coupled Schrödinger system with a Hardy potential in the critical case

where $2^*=\frac{2N}{N-2}$, $\lambda_i\in (0,\Lambda_N), \Lambda_N:= \frac{(N-2)^2}{4}$, and $\beta_{ij}=\beta_{ji}$ for i ≠ j. By virtue of variational methods, we establish the existence and nonexistence of least energy solutions for the purely cooperative case ($\beta_{ij} \gt 0$ for any i ≠ j) and the simultaneous cooperation and competition case ($\beta_{i_{1}j_{1}} \gt 0$ and $\beta_{i_{2}j_{2}} \lt 0$ for some $(i_{1}, j_{1})$ and $(i_{2}, j_{2})$). Moreover, it is shown that fully nontrivial ground state solutions exist when $\beta_{ij}\ge0$ and $N\ge5$, but NOT in the weakly pure cooperative case ($\beta_{ij} \gt 0$ and small, i ≠ j) when $N=3,4$. We emphasize that this reveals that the existence of ground state solutions differs dramatically between $N=3, 4$ and higher dimensions $N\geq 5$. In particular, the cases of N = 3 and $N\geq 5$ are more complicated than the case of N = 4 and the proofs heavily depend on the dimension. Some novel tricks are introduced for N = 3 and $N\ge5$.