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Let $r$ be the number of conjugacy classes in $G$. The action of $G$ on itself by conjugation gives, via the Cauchy-Frobenius formula, $r=\frac{1}{|G|}\sum |C(g)|$, where $g$ ranges over $G$. This action restricted to $H$ gives the number of $G$-conjugacy classes in $H$ as $\frac{1}{|G|}\sum |C_H(g)|$, and from the fact $|C_H(g)|\ge \frac{1}{k}|C(g)|$ the first part of your problem follows.

I will continue to think about

EDIT: As pointed out by Sergei Ivanov below, this argument also shows the second part is true as well.

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Let $r$ be the number of conjugacy classes in $G$. The action of $G$ on itself by conjugation gives, via the Cauchy-Frobenius formula, $r=\frac{1}{|G|}\sum |C(g)|$, where $g$ ranges over $G$. This action restricted to $H$ gives the number of $G$-conjugacy classes in $H$ as $\frac{1}{|G|}\sum |C_H(g)|$, and from the fact $|C_H(g)|\ge \frac{1}{k}|C(g)|$ the first part of your problem follows.

I will continue to think about the second part.