There is a famous result of Banyaga stating that if two closed symplectic manifolds $(M_1, \omega_1)$ and $(M_2, \omega_2)$ have isomorphic groups of Hamiltonian diffeomorphisms $\mathrm{Ham}(M_1, \omega_1)\simeq \mathrm{Ham}(M_2, \omega_2)$ then there exists a diffeomorphism $f:M_1\rightarrow M_2$ such that $f^*\omega_2=\lambda \omega_1$ for some $\lambda\in\mathbb{R}^{*}$. In other words, a symplectic structure on a closed manifold is determined (up to rescaling) by the group of Hamiltonian diffeomorphisms.

The result of Banyaga tells us that, in principle, it should be possible to understand the topology of $M$ only from $\mathrm{Ham}(M, \omega)$. However, as far as I understand, it's pretty hard to actually give an algorithm for doing that.

So my question is: are there any explicit procedures for recovery of algebro-topological invariants of $M$ from $\mathrm{Ham}(M, \omega)$? I am particularly interested in recovery of the fundamental group $\pi_1(M)$.

A weaker question would be: is there any explicit condition on $\mathrm{Ham}(M, \omega)$ which guarantees that $M$ is simply-connected?

I am aware of one comparatively weak result in this direction. Banyaga has constructed a map (called flux map) $f:\pi_1(\mathrm{Symp}_0(M, \omega))\rightarrow H^1(M, \mathbb{R})$. He has also shown that there is an isomorphism $$ \mathrm{Symp}_0(M, \omega)/\mathrm{Ham}(M, \omega) \simeq H^1(M, \mathbb{R})/\mathrm{ker}\:f. $$ Therefore, if we know both $\mathrm{Symp}^0(M, \omega)$ and $\mathrm{Ham}(M, \omega)$ we can at least recover a quotient of $H^1(M, \mathbb{R})$ (so we have a lower bound for its rank, for example).

P.S.: A clear exposition of some results on the group of Hamiltonian diffeomorphisms can be found in Polterovich's book "The geometry of the group of symplectic diffeomorphisms" (chapter 14 is particularly relevant to the question).