= Solution
<Moser's trick> states that if $M$ is compact and $\omega_t$, $0\leq t\leq1$, is a smooth family of <symplectic forms> whose <de Rham cohomology> class is independent of $t$, then there is an <isotopy> $\phi_t$ with
$$
\phi_t^*\omega_t=\omega_0.
$$
Since $[\dot\omega_t]=0$, choose a smooth family of one-forms $\sigma_t$ with $\dot\omega_t=d\sigma_t$. Nondegeneracy of $\omega_t$ uniquely determines a vector field $X_t$ by
$$
\iota_{X_t}\omega_t=-\sigma_t.
$$
Compactness makes its flow $\phi_t$ exist for the whole interval. <Cartan's magic formula> and $d\omega_t=0$ give
$$
\frac d{dt}\phi_t^*\omega_t
=\phi_t^*(\dot\omega_t+\mathcal L_{X_t}\omega_t)
=\phi_t^*(d\sigma_t+d\iota_{X_t}\omega_t)=0,
$$
which proves the theorem.
Smooth degree-$d$ hypersurfaces form the complement of the discriminant in the projective space of degree-$d$ homogeneous polynomials. This complement is path connected, so $X$ and $X'$ lie in a smooth one-parameter family. The <Ehresmann fibration theorem> identifies the fibers smoothly. Under such an identification, the restrictions of the <Fubini-Study form> form a family $\omega_t$ whose cohomology class is the fixed restricted hyperplane class. Moser's trick therefore gives the <symplectic equivalence of smooth projective hypersurfaces>.
It remains to construct the finite subgroup for one convenient hypersurface. On the <Fermat hypersurface>
$$
X_F=\{z_0^d+\cdots+z_n^d=0\}\subset\mathbb{CP}^n,
$$
the group $(\mu_d)^{n+1}$ acts by diagonal coordinate multiplication. It preserves both $X_F$ and the Fubini-Study form. The kernel of its projective action is the diagonal subgroup $\mu_d$, so the effective <Fermat hypersurface diagonal symmetry> group is
$$
(\mathbb Z/d\mathbb Z)^{n+1}/\langle(1,\ldots,1)\rangle
\cong(\mathbb Z/d\mathbb Z)^n.
$$
Conjugating this action by a symplectomorphism $X_F\to X$ gives the required subgroup of $\operatorname{Symp}(X,\omega_{FS}|_X)$.
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