I see above counter-examples in this thread. On the other hand, in the Abelian case there are positive results, which go way further than the question. In the case of (general) solenoids, they themselves are their own first homology groups with coefficients in $S^1$. It follows that when two of them are homeomorphic (or just homologically equivalent) then they are isomorphic as topological groups--see Karol Borsuk and myself, Fund. Math. 1970. This line was developed much further in later papers by James Keesling.

I see above counter-examples in this thread. On the other hand, in the Abelian case there are positive results, which go way further than the question. In the case of (general) solenoids, they themselves are their own first homology groups with coefficients in $S^1$. It follows that when two of them are homeomorphic (or just homologically equivalent) then they are isomorphic as topological groups--see Karol Borsuk and myself, Fund. Math. 1970. This line was developed much further in later papers by James Keesling.

The following five classifications of (general) solenoids are identical: (i) homological (with coefficients in $S^1$; i.e. cohomological with coefficients in $Z$; for Čech theories), (ii) shape-theoretical, (iii) homotopic, (iv) topological, (v) as topological groups.