Let $G$ be the set of finitely generated groups up to isomorphism.
Now define $B$ and $C$ two finitely generated groups to be not discernable if one can find a finitely generated group $A$ such that $A\times B$ is isomorphic to $A\times C$. This defines an equivalence relation on $G$ (straightforward verification).
The questions I will ask are related to this post :
In this post, from the counter-example for $(1)$ not every class is reduced to one (you can not always discern a group from another). From the counter-example to $(2)$ there are many classes (using the abelianization one can discern groups using edit 2). So the "non-discernability" is clearly a non-trivial relation (so this is different from both the finite groups case and the general groups case). I recall that we are in $G$ and hence all groups are finitely generated.
(3) Are there groups $B$ such that the "non-discernability" class is reduced to $B$ ? If yes, can we caracterize them ?
(4) Is the trivial group "non-discernability" class reduced to the trivial group? If not, can we caracterize groups within this class ?
($\infty$) Give invariants for the "non-discernability" classes.