(2) For which positive integers $k$, does the inequality
$$a^{k}+b^{k}+c^{k}+d^{k}-k a b c d \geqslant M_{k}$$

hold for all real numbers $a, b, c, d$? Find the maximum possible value of $M_{k}$, and indicate when equality holds.
(2) Notice that, when $k$ is odd, choosing sufficiently large negative values for $a, b, c, d$ results in a sufficiently large negative value for $a^{k}+b^{k}+c^{k}+d^{k}-k a b c d$. Therefore, such a number $M_{k}$ does not exist.

When $k=2$, taking $a=b=c=d=r$, we get $a^{2}+b^{2}+c^{2}+d^{2}-2 a b c d=4 r^{2}-2 r^{4}$.

Choosing sufficiently large positive values for $r$ also yields arbitrarily large negative values. Therefore, such a number $M_{k}$ does not exist.

When $k$ is even and $k \geqslant 4$, taking $a=b=c=d=1$, we get $a^{k}+b^{k}+c^{k}+d^{k}-k a b c d=4-k$.

Similarly to (1), we have
$$a^{k}+b^{k}+c^{k}+d^{k}-k a b c d \geqslant 4-k$$

That is, $\frac{a^{k}+b^{k}+c^{k}+d^{k}+(k-4) \cdot 1^{k}}{k} \geqslant a b c d$.
The equality holds under the same conditions as in (1).
Thus, the maximum value of $M_{k}$ in this case is $4-k$.