Let us say a group $(G,\cdot)$ has character $0$ if for all $g\in G\setminus\{e\}$ and for all positive integers $n$ we have $g^n \neq e$ (where $e$ is the neutral element of the group.

Is there a collection ${\cal G}$ of countably infinite groups, such that

1. each $G\in {\cal G}$ has character $0$,
2. if $G_1\neq G_2\in{\cal G}$ then $G_1\not\cong G_2$, and
3. $|{\cal G}| = 2^{\aleph_0}$?

?