= Solution
The body is invariant under the reflections and half-turns that preserve its $z$-axis. A polar vector force $F\mathbf e_z$ can therefore produce only the polar velocity $U_z\mathbf e_z$; every rotation is excluded because <angular velocity> is a <pseudovector>. Hence only $U_z$ is nonzero when $\mathbf F=(0,0,F)$ and $\mathbf G=0$.
For $\mathbf F=(F,0,0)$, the surviving symmetry-allowed components are
$$
\boxed{U_x\ne0,\qquad\Omega_y\ne0},
$$
while $U_y,U_z,\Omega_x,\Omega_z$ vanish. The coupling between translation in $x$ and rotation about $y$ is permitted because the two horizontal rods lie at opposite vertical offsets.
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