  The decay $\overline{B}_s^0 \rightarrow \psi(2S) K^+ \pi^-$ is observed using
a data set corresponding to an integrated luminosity of $3.0fb^{-1}$ collected
by the LHCb experiment in $pp$ collisions at centre-of-mass energies of 7 and 8
TeV. The branching fraction relative to the $B^0\rightarrow \psi(2S) K^+ \pi^-$
decay mode is measured to be \begin{equation} \frac{{\cal B}(\overline{B}^0_s
\rightarrow \psi(2S) K^+ \pi^-)}{{\cal B}(B^0
  \rightarrow \psi(2S) K^+ \pi^-)} = 5.38 \pm 0.36 (stat) \pm 0.22 (syst) \pm
0.31 \, (f_s/f_d) \, \%,\nonumber \end{equation} where $f_s/f_d$ indicates the
uncertainty due to the ratio of probabilities for a $b$ quark to hadronise into
a $B_s^0$ or $B^0$ meson. Using an amplitude analysis, the fraction of decays
proceeding via an intermediate $K^*(892)^0$ meson is measured to be $0.645 \pm
0.049 (stat) \pm 0.049 (syst)$ and its longitudinal polarisation fraction is
$0.524 \pm 0.056 (stat) \pm 0.029 (syst)$. The relative branching fraction for
this component is determined to be \begin{equation} \frac{{\cal
B}(\overline{B}^0_s \rightarrow \psi(2S) K^*(892)^0)}{{\cal B}(B^0
  \rightarrow \psi(2S) K^*(892)^0)} = 5.58 \pm 0.57 (stat) \pm 0.40 (syst) \pm
0.32 \, (f_s/f_d) \, \%. \nonumber \end{equation} In addition, the mass
splitting between the $B_s^0$ and $B^0$ mesons is measured as \begin{equation}
M(B^0_s) - M(B^0) = 87.45 \pm 0.44 (stat) \pm 0.07 (syst) MeV/c^2. \nonumber
\end{equation}
