Finite groups are 1-isomorphic iff their directed power graphs are isomorphic

Statement
Suppose $$G$$ and $$H$$ are fact about::finite groups. Then, $$G$$ and $$H$$ are fact about::1-isomorphic groups (in particular, fact about::1-isomorphic finite groupsif and only if their directed power graphs are isomorphic as directed graphs.

Related facts

 * Undirected power graph determines directed power graph for finite group
 * Undirected power graph need not determine directed power graph for infinite group
 * Directed power graph-equivalent not implies 1-isomorphic for infinite groups