= Solution
Choose the phase of each inward-pointing ligand <sigma bond> donor lobe consistently. An operation taking one donor to another then acts as a permutation on the four donor functions. For the square, label these functions $L_1,L_2,L_3,L_4$ cyclically along $+x,+y,-x,-y$. Normalized <symmetry-adapted linear combinations> are
$$
\begin{aligned}
A_{1g}:&\quad(L_1+L_2+L_3+L_4)/2,\\
B_{1g}:&\quad(L_1-L_2+L_3-L_4)/2,\\
E_u:&\quad(L_1-L_3)/\sqrt2,\quad(L_2-L_4)/\sqrt2.
\end{aligned}
$$
Thus the ligand <group representation> is \b[$\Gamma_\sigma=A_{1g}+B_{1g}+E_u$]. Its <irreducible representations> allow the first two combinations to mix with the matching nickel $d$ <atomic orbitals>; $E_u$ instead matches nickel $p_x,p_y$.
For the tetrahedron in the vertex order specified above, take
$$
\begin{aligned}
A_1:&\quad(L_1+L_2+L_3+L_4)/2,\\
T_2:&\quad(L_1+L_2-L_3-L_4)/2,\\
&\quad(L_1-L_2+L_3-L_4)/2,\\
&\quad(L_1-L_2-L_3+L_4)/2.
\end{aligned}
$$
The three latter functions transform like $x,y,z$. Hence \b[$\Gamma_\sigma=A_1+T_2$]. These are four ligand <symmetry-adapted linear combinations>, rather than the five metal $d$ functions of part (ii).
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