An isomorphism requires a metric, which metric do you mean?

I'm talking group homomorphic, though they're also topologically isomorphic, a.k.a. homeomorphic.

Saying that two groups are homomorphic doesn't tell me much more than there exists a group homomorphism... As far as I know, isomorphism implies one can be transformed into the other through renaming.

I don't know anything about topology, so I'll take your word for it.