Let $G$ be an arbitrary fixed group.
First, we know that if $f: G \rightarrow G'$ is a group homomorphism, then $\ker f$ is a normal subgruop of $G$.
Second, we also know that if $H$ is a normal subgroup of $G$, then the canonical map from $G$ into $G/H$ is a homomorphism whose kernel is $H$.
If $f_1: G \rightarrow G_1$ and $f_2: G \rightarrow G_2$ are two surjective group homomorphisms whose kernel are same, then is there an isomorphism $h:G_1 \rightarrow G_2$ such that $h \circ f_1 = f_2$?
If the answer for the above question is true, then can we say that "the cardiality of surjective group homomorphism whose domain is $G$ (up to isomorphism) is equal to the cardinality of the set of normal subgroups of $G$"?
Is there more relationships between the set of surjective homomorphisms (from $G$) and the set of normal subgroups of $G$?
Thank you for concerning about this question.