gr.group theory - Abelianization of a semidirect product

I agree with Ryan and Victor, except that you don't need presentations. The subgroup $[G ltimes H,G ltimes H]$ is generated by $[H,H] cup [G,H] cup [G,G]$, so you can write
$$(G ltimes H)^{ab} = (G ltimes H) / langle [H,H] cup [G,H] cup [G,G] rangle.$$
If you apply the relators $[H,H]$, you get $G ltimes H^{ab}$; then if you apply the relators $[G,H]$, you get $G times (H^{ab})_G$; then finally if you apply $[G,G]$, you get $G^{ab} times (H^{ab})_G$. You can add this as an extra half-paragraph or footnote rather than giving a citation.

I don't think that the referee has the right to demand a longer explanation than this, unless maybe you are writing a textbook.

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