Is there a natural family of nonisomorphic groups parametrized by $\mathbb{R}$?

For $s\in\mathbf{R}$ consider the group $G_s$ of matrices $$\begin{pmatrix}e^t & 0 & x\\ 0 & e^{st} & y\\ 0 & 0 & 1\end{pmatrix}: \quad x,y,t\in\mathbf{R}.$$

This is a 3-dimensional connected Lie group. Then $G_s$ and $G_t$ are isomorphic Lie groups if and only if $st=1$ (exercise). Hence for $s>1$ they're pairwise non-isomorphic.

As proved in this question, the groups $G_{x^r}$ for $r\in (0,1]$ (defined as permutations of $\mathbb{N}$ that fix all but $o(n^r)$ elements) are pairwise non-isomorphic.