= Solution
The nine-atom nonlinear ion has \b[$3N-6=21$ vibrational degrees of freedom]. To construct its displacement <group representation>, only atoms left in place by an operation contribute to the <character>. Each such atom contributes the trace of the operation on $(x,y,z)$:
$$
\chi_{3N}(R)=N_{\rm fixed}(R)\,\operatorname{tr}D_{xyz}(R).
$$
For example, reflection in the square's molecular plane fixes all nine atoms and has trace $1$, giving character $9$; inversion fixes nickel alone and has trace $-3$, giving $-3$. <Molecular character reduction> uses
$$
n_\lambda=\frac1{|G|}\sum_C |C|\,\chi_{3N}(C)\chi_\lambda(C)^*.
$$
For the <D4h molecular point group>, with primed twofold axes along the Ni–C directions, the class order and displacement characters are
$$
\begin{array}{c|rrrrrrrrrr}
C&E&2C_4&C_2&2C_2'&2C_2''&i&2S_4&\sigma_h&2\sigma_v&2\sigma_d\\\hline
\chi_{3N}&27&1&-1&-5&-1&-3&-1&9&5&1
\end{array}
$$
Reducing and removing translations $A_{2u}+E_u$ and rotations $A_{2g}+E_g$ gives
$$
\boxed{\Gamma_{\rm vib}^{D_{4h}}
=2A_{1g}+A_{2g}+2B_{1g}+2B_{2g}+E_g
+2A_{2u}+2B_{2u}+4E_u.}
$$
The <vibrational infrared activity> belongs to $2A_{2u}+4E_u$, giving \b[six features]. The <Raman vibrational activity> belongs to $2A_{1g}+2B_{1g}+2B_{2g}+E_g$, giving \b[seven features]. The remaining $A_{2g}+2B_{2u}$ modes are silent. There are \b[zero shared fundamentals], by <infrared and Raman mutual exclusion>. Counting dimensions gives $10$ IR-active, $8$ Raman-active and $3$ silent vibrational coordinates, totaling $21$.
For the <Td molecular point group>, the corresponding calculation starts from
$$
\begin{array}{c|rrrrr}
C&E&8C_3&3C_2&6S_4&6\sigma_d\\\hline
\chi_{3N}&27&0&-1&-1&5
\end{array}
$$
and gives $\Gamma_{3N}=2A_1+2E+2T_1+5T_2$. Subtracting translations $T_2$ and rotations $T_1$ leaves
$$
\boxed{\Gamma_{\rm vib}^{T_d}=2A_1+2E+T_1+4T_2.}
$$
The $4T_2$ fundamentals give \b[four IR features]. The $2A_1+2E+4T_2$ fundamentals give \b[eight Raman features], of which \b[four coincide with the IR features]. The $T_1$ fundamental is silent. The dimensions are $2+4+3+12=21$.
Thus the predicted IR/Raman/coincidence counts are \b[$6/7/0$ for the square and $4/8/4$ for the tetrahedron]. These are numbers of distinct ideal fundamental frequencies: each $E$ or $T$ multiplet contributes one feature, and different copies of the same <irreducible representation> generally have different frequencies. Accidental degeneracy, overtones and combination bands are outside this count.
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