  We study the nonlinear Schr\"odinger equation with $\delta'_s$ coupling of
intensity $\beta\in\mathbb{R}\setminus\{0\}$ on the star graph $\Gamma$
consisting of $N$ half-lines. The nonlinearity has the form $g(u)=|u|^{p-1}u,
p>1.$ In the first part of the paper, under certain restriction on $\beta$, we
prove the existence of the ground state solution as a minimizer of the action
functional $S_\omega$ on the Nehari manifold. It appears that the family of
critical points which contains a ground state consists of $N$ profiles (one
symmetric and $N-1$ asymmetric). In particular, for the attractive $\delta'_s$
coupling ($\beta<0$) and the frequency $\omega$ above a certain threshold, we
managed to specify the ground state.
  The second part is devoted to the study of orbital instability of the
critical points. We prove spectral instability of the critical points using
Grillakis/Jones Instability Theorem. Then orbital instability for $p>2$ follows
from the fact that data-solution mapping associated with the equation is of
class $C^2$ in $H^1(\Gamma)$. Moreover, for $p>5$ we complete and concertize
instability results showing strong instability (by blow up in finite time) for
the particular critical points.
