= Finite extensions preserve finite pro-p presentations
Suppose $K$ is an open <normal subgroup> of a <pro-p group> $G$ and has a <finite pro-p presentation>. Choose lifts of generators of the finite quotient $G/K$. Add finitely many relators giving their conjugation and inverse conjugation on the generators of $K$, and replace each quotient relator by its value in $K$. The resulting finite <finite pro-p presentation> maps onto $G$. Its closed subgroup generated by the kernel generators is isomorphic to $K$, and its quotient by that subgroup is exactly $G/K$, so the map is also injective.
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