Given a finite group $G$, let $1,n_1,n_2,\cdots, n_k$ denote all the possible sizes of conjugacy classes of $G$, with $1<n_1<n_2\cdots$. The first remarkable theorem by concerning such sequence was obtained by Noboru Ito (~ 1954, and later):
If $G$ is a finite group and if $\{1,n_1\}$ are the only possible conjugacy class sizes ($n_1>1$) then $G$ is nilpotent (and something more....).
If $G$ is a finite group and if $\{1,n_1,n_2\}$ are the only possible conjugacy class sizes then $G$ is solvable.
After this result, many variations were done in it, as well as imposing related questions, different results obtained. In a recent paper (2008), Avinoam Mann says
It should be noted that these results pre-date the classification of finite simple groups, and possibly were motivated by the quest for that classification.
Question: Is this true that the above results of Ito (or its variations) are used in the classification of finite simple groups? If yes, for what kind of question (or for which types of groups) they were used?