We study the symplectomorphism groups $G_��=Symp_0(M,��_��)$ of an arbitrary closed manifold M equipped with a 1-parameter family of symplectic forms $��_��$ with variable cohomology class. We show that the existence of nontrivial elements in $��_*({\cal A},{\cal A}')$, where $({\cal A},{\cal A}')$ is a suitable pair of spaces of almost complex structures, implies the exiarxiv.org stence of families of nontrivial elements in $��_{*-i}G_��$, for $i=1$ or 2. Suitable parametric Gromov Witten invariants detect nontrivial elements in $��_*({\cal A},{\cal A}')$. By looking at certain resolutions of quotient singularities we investigate the situation $(M,��_��)= (S^2 \times S^2 \times X,��_F \oplus ����_B \oplus ��_{st})$, with $(X,��_{st})$ an arbitrary symplectic manifold. We find families of nontrivial elements in $��_k(G_��^X)$, for countably many $k$ and different values of $��$. In particular we show that the fragile elements $w_{\ell}$ found by Abreu-McDuff in $��_{4 \ell}(G_{\ell+1}^{pt})$ do not disappear when we consider them in $S^2 \times S^2 \times X$.

23 pages