Let $G$ be a finitely generated group. A random subgroup $\mu$ of $G$ is a Borel probability measure on the space $\text{Sub}(G)$ of subgroups of $G$. The topology on $\text{Sub}(G)$ is the one induced by its inclusion in $\{0,1\}^G$, where the latter space is endowed with the product topology (and $\{0,1\}$ with the discrete topology). We say that $G$ is an invariant random subgroup if $\mu(gAg^{-1})=\mu(A)$ for every Borel subset $A$ of $\text{Sub}(G)$ and $g\in G$.
One way to generate an invariant random subgroup of $G$ is to take a subgroup $H$ of $G$ which has only a finite number $c$ of conjugates in $G$, and let $\mu$ be the uniform measure assigning measure $1/c$ to each conjugate. More generally, we could take countable convex combinations of invariant random subgroups of this form.
The measures described above are just countable convex combinations of Dirac measures.
I am looking for examples of invariant random subgroups on a finitely generated group $G$ which are not countable convex combinations of Dirac measures.
I mainly care about the case where $G$ is a finitely generated free group. It would be great to have many examples.
Take a finitely generated group $G$ with a center $Z$ having continuum many subgroups (e.g., $Z$ contains $\mathbf{Z}[1/p]^2$ for some prime $p$). Then any non-atomic measure on the set of subgroups of $Z$ works.