The MathSciNet review of the book Berkovich, Zhmud, Characters of finite groups. Part 2, says the following:

...Let $k(G)$ be the number of conjugacy classes of the group $G$, $T(G)$ the sum of the degrees of its irreducible characters and $t(G)$ the number of its involutions. Chapter 28 characterizes the groups with $k(G)\ge 1/16|G|$, the groups with $T(G)\ge 1/4|G|$ and the groups with $t(G)\ge 1/4|G|$....

Since I cannot get a hold of the book soon, I would appreciate if someone could briefly list these results, and/or give me the original references where these problems are addressed.