I'm interested in the following situation:

- $G$ is a finite group;
- $C$ is a conjugacy class in $G$;
- $H$ is the centralizer of an element $h$ of $C$.

I want information on $|C\cap Hg|$ as $g$ varies across $G$. In particular I'd like to prove that there exists $k<1$ such that for all $g\in G$ we have $$|C\cap Hg| \leq k|H|.$$

Unfortunately for me such a bound does not exist in complete generality: consider $C_p\rtimes C_{p-1}$ for a prime $p$ (semidirect product of two cyclic groups). Let $C$ be the conjugacy class of elements of order $p$, all of which have the same centralizer $H$. Then $C$ is a subset of $H$ and we have $$|C\cap H| = (p-1/p)|H|.$$ So as $p$ goes to infinity we have $(p-1/p)\to 1$.

So we can only prove a bound of the given form for particular cases. With this in mind here are some questions:

- Is it true that $|C\cap Hg|\leq |C\cap H|$?
**Edit:**No it is not true. Mark Wildon has provided counter-examples in his answer below. If we assume that $G$ is simple does a bound of the given form with $k<1$ exist? - Does anyone know if this problem appears in the literature in an alternative formulation? I'm interested even in particular cases, e.g. taking G to be a particular family of simple groups and C a particular family of conjugacy classes.
**Edit:**As discussed in comments below, the case when $|C\cap Hg|=1$ for all $g\in G$ corresponds precisely to the situation $G=HC$. An example of this phenomenon is given below when $G=C_p\rtimes C_{p-1}$, a Frobenius group. Does this ever happen for $G$ simple? Has the problem of decomposing a group $G$ into the product of a centralizer and conjugacy class been studied in the literature?