Actions of $\mathbb{F}_\infty$ whose II$_1$ factors and orbit equivalence relations have prescribed fundamental group

We show that given any subgroup $\mathcal{F}$ of $\mathbb{R}_+$ which is either countable or belongs to a certain ``large'' class of uncountable subgroups, there exist continuously many free ergodic measure-preserving actions $\sigma_i$ of the free group with infinitely many generators $\mathbb{F}_\infty$ on probability measure spaces $(X_i,\mu_i)$ such that their associated group measure space II$_1$ factors $M_i=\operatorname{L}^\infty(X_i) \rtimes_{\sigma_i} \mathbb{F}_\infty$ and orbit equivalence relations $\mathcal{R}_i=\mathcal{R} (\mathbb{F}_\infty {\overset{}{\curvearrowright}} X_i)$ have fundamental group equal to $\mathcal{F}$ and with $M_i$ (respectively $\mathcal {R}_i$) stably non-isomorphic. Moreover, these actions can be taken so that $\mathcal{R}_i$ has no outer automorphisms and any automorphism of $M_i$ is unitarily conjugate to an automorphism that acts trivially on the subalgebra $\operatorname{L}^\infty(X_i)$ of $M_i$.