 Finite presentation of a semidirect product

If $N=⟨A\mid R⟩$ and $H=⟨B\mid S⟩$ have finite group presentations and $\varphi :H\to \mathrm{Aut}(N)$ is an action, choose a word $w_{b,a}(A)$ representing $\varphi (b)(a)$. Then $N{⋊}_{\varphi }H$ has the finite presentationThe 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 $R$.