In Theorem 3.1 in this paper, part (1) of the proof, $X$ may not be a subgroup of $N$, so how did the authors apply the induction method?
Also, I do not understand this: In the proof of Theorem 3.2 he applied Theorem 3.1 with the possibility that $X$ may not be a subgroup of $H$.
This is a link for the paper in reference [3] which has the proof of Lemma 2.1.