= Finite presentation of a semidirect product
If $N=\langle A\mid R\rangle$ and $H=\langle B\mid S\rangle$ have <finite group presentations> and $\phi:H\to\operatorname{Aut}(N)$ is an action, choose a word $w_{b,a}(A)$ representing $\phi(b)(a)$. Then $N\rtimes_\phi H$ has the finite presentation
$$
\langle A,B\mid R,S,\ bab^{-1}=w_{b,a}(A)\ (a\in A,b\in B)\rangle.
$$
The action automorphisms also let these relations rewrite conjugation by inverse generators. Every word becomes an $A$-word followed by a $B$-word. The natural map to the <semidirect product> is injective because its triviality forces the $B$-word to be trivial using $S$ and then the $A$-word to be trivial using $R$.
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