Conjugacy limits of certain subgroups in $\text{SL}(2,\mathbb{R})⋉{\mathbb{R}}^2$

We study conjugacy limits of certain of subgroups inside $\text{SL}(2,\mathbb{R})⋉{\mathbb{R}}^2$. These subgroups have a common feature that any two in the same category are conjugates of each other.