Does there exist two abelian groups $A,B$ with an epimorphism $f: A\to B$, and two other abelian groups $A', B'$ along with an epimorphism $g: A'\to B'$ such that $A\cong A'$, $B\cong B'$ and $ker\,f \not\cong ker\,g$? It seems to me that the groups must be infinite, since we have $B\cong A/ker\,f$ and $B'\cong A'/ker\,g$.
Thanks!