In these notes, http://sporadic.stanford.edu/bump/group/gr2_2.html, I don't understand why there's a need for Proposition 2.2.1 in view of Lemma 2.2.3 (which comes before). I thought Lemma 2.2.3 says "if we have a character $\chi $ on a proper subgroup $H$ of $G$, then we can extend $\chi $ to a group $J$ such that $H\subset J\subseteq G$". (All groups here are finite (and abelian) by the way).
But maybe I'm interpreting this wrong and getting confused about largest/maximal sets or something like this? Why can't we just keep applying the above lemma until we reach $G$?