= Solution
The <pushout of groups> is
$$
G=(G_1*G_2)/\left\langle\!\left\langle i_1(h)i_2(h)^{-1}:h\in H\right\rangle\!\right\rangle.
$$
Its maps from $G_1,G_2$ agree on $H$, and every other pair of <group homomorphisms> agreeing on $H$ factors through one unique homomorphism from $G$. This is the <universal property>. When the $i_j$ are injective, it is the <amalgamated free product>, and the two factors and the identified copy of $H$ embed in it. Some conventions use this name for the quotient even without injectivity; the embedding assertion requires injectivity.
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