Example of $2$ non-isomorphic groups that have the same quotients I'm looking for an example of $2$ non isomorphic groups $G_1,G_2$  that are finitely generated and presented that have the same finite quotients (up to isomorphism) .

Thanks
In my answer to this question, I wrote down finite presentations from a paper by Baumslag and Solitar of two non-isomorphic groups that are each isomorphic to a quotient group of the other. So these two groups have exactly  the same quotient groups, not just the same finite quotient groups.
Let $Z_k$ be the cyclic group of order $k$.

Consider, $A = Z_2 \times Z_2$ and $B = Z_4$.

These are not isomorphic to each other and the only non-trivial quotient they have is $Z_2$.