  The energy-dependent cross section for $e^+e^-\to \eta\psi(2S)$ is measured
at eighteen center of mass energies from 4.288 GeV to 4.951 GeV using the
BESIII detector. Using the same data samples, we also perform the first search
for the reaction $e^+e^-\to\eta\tilde{X}(3872)$, but no evidence is found for
the $\tilde{X}(3872)$ in the $\pi^+\pi^- J/\psi$ mass distribution. At each of
the eighteen center of mass energies, upper limits at the 90\% confidence level
on the cross section for $e^+e^-\to\eta\psi(2S)$ and on the product of the
$e^+e^-\to\eta\tilde{X}(3872)$ cross section with the branching fraction of
$\tilde{X}(3872)\to\pi^+\pi^- J/\psi$ are reported.
