Directed power graph-equivalent not implies 1-isomorphic for infinite groups

Statement

It is possible to have two infinite groups $G$ and $H$ that are directed power graph-equivalent (i.e., they have isomorphic directed power graphs) and are not 1-isomorphic groups (?).

Related facts

• Finite groups are 1-isomorphic iff their directed power graphs are isomorphic
• Undirected power graph determines directed power graph for finite group

Proof

For any prime $p$, consider the group $G_{p}$ of rational numbers that, in reduced form, have denominator a power of $p$. The isomorphism class of the directed power graph of $G_{p}$ is independent of $p$.