Directed power graph-equivalent groups

Definition
Two groups $$G$$ and $$H$$ are directed power graph-equivalent groups if the directed power graph of $$G$$ is isomorphic (as a directed graph) to the directed power graph of $$H$$.

Finite version
If either of the groups $$G$$ or $$H$$ is a finite group, so is the other group, and in this case they are both 1-isomorphic finite groups.

Facts

 * Finite groups are 1-isomorphic iff their directed power graphs are isomorphic
 * Directed power graph-equivalent not implies 1-isomorphic for infinite groups
 * Undirected power graph determines directed power graph for finite group