= Solution
One <crystallographic inversion center> is
$$
\boxed{(x_0,y_0,z_0)=(\tfrac14,\tfrac14,0).}
$$
The inversion maps $(x,y,z)$ to $(1/2-x,1/2-y,-z)$; it exchanges both the Fe pair and the Se pair modulo <lattice translations>.
A $\bar4$ axis passes along $z$ through the Fe site $(0,0,0)$, and likewise through $(1/2,1/2,0)$. A representative <rotoinversion> is
$$
\boxed{(x,y,z)\mapsto(y,-x,-z).}
$$
It leaves the first Fe site fixed, maps the other Fe onto a translationally equivalent site, and exchanges the Se height classes. The $\bar4$ operation includes a vertical reversal, so it must not be replaced by a pure $90$-degree rotation about an Fe site. Such a pure rotation would put a selenium atom at the wrong height.
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