Question. Can a countable group have an uncountable number of distinct Hausdorff group topologies?
By a group topology one understands a topology with respect to which the group operations are continuous. By distinct I mean non-isomorphic as topological groups.
Motivation. The examples I have in mind are the pro-$p$ topologies on (e.g.) free groups, which typically yield an infinite (but countable) number of distinct topologies. We can take the uncountably many subsets $\varpi$ of the primes and look at the pro-$\varpi$ topologies, but I'm not certain that they are all distinct.
(Interestingly, it seems that a kind of dual question has been asked before, which popped up in the list of similar questions.)