FALSE: canonical factor maps into every group pushout are injective

Given: The objects and hypotheses in the statement.

For a pushout of $f:K\to G$ and $h:K\to H$, $i_H(h(\ker f))=\{e\},\qquad i_G(f(\ker h))=\{e\}.$ Hence canonical factor maps in an arbitrary group pushout need not be injective. No equality with their full kernels is asserted. (The kernels of the amalgamating maps are killed in the opposite canonical maps to a group pushout).

For homomorphisms $f:K\to G$ and $h:K\to H$, let $N$ be the normal closure in $G\ast H$ of $\{j_G(f(k))j_H(h(k))^{-1}:k\in K\}.$ Then $(G\ast H)/N$, with the induced factor maps $j_G$ and $j_H$, is a pushout of $f$ and $h$. (A group pushout is the quotient of a free product by the amalgamating relations).

For every $n\in\mathbb N$, view $n$ as its canonical nonnegative integer and put $n\mathbb Z:=\{nk:k\in\mathbb Z\}$. Then the left cosets of $n\mathbb Z$ in $(\mathbb Z,+)$ are exactly the congruence classes modulo $n$, and coset addition is the published addition of congruence classes. Thus $(\mathbb Z,+)/n\mathbb Z=(\mathbb Z/n,+)$ as the same group on the same underlying set. This includes $n=0$ and $n=1$. (For every $n\in\mathbb N$, the congruence-class group $(\mathbb Z/n,+)$ is the quotient group $(\mathbb Z,+)/n\mathbb Z$).