Undirected power graph need not determine directed power graph for infinite group

Statement
It is possible to have infinite groups $$G$$ and $$H$$ such that $$G$$ and $$H$$ have isomorphic undirected power graphs but do not have isomorphic directed power graphs. In particular, they need not be fact about::1-isomorphic groups.

Proof
For any prime number $$p$$, the $$p$$-quasicyclic group is, up to isomorphism, the group of the union of all $$(p^n)^{th}$$ roots of unity for all nonnegative integers $$n$$, under multiplication of complex numbers. Equivalently, it is the direct limit of a sequence of groups of order $$p^n$$ with each group injecting into the next one naturally.

The undirected power graph of any $$p$$-quasicyclic group is a countable complete graph. Hence, the undirected power graphs of $$p$$-quasicyclic groups for different primes $$p$$ are isomorphic. On the other hand, the directed power graphs are not isomorphic, because we can use the directed power graph to determine whether an element has order $$p$$ (by counting the edges outward from it) and hence use this to distinguish between $$p$$-quasicyclic groups for different $$p$$.