I am trying to recall what happened to the conjugacy class and centraliser when we quotient out by a normal subgroup.
In particular, we know from Orbit-Stabiliser that
$|G|=|ccl_G(g)||C_G(g)|$ for every $g\in G$
If we quotient out by a normal subgroup we have that
$|G|/|N|=|G/N|=|ccl_{G/N}(gN)||C_{G/N}(gN)|$
My question is what was the relationship between $|ccl_G(g)|$ and $|ccl_{G/N}(gN)|$ (or $|C_G(g)|$ and $|C_{G/N}(gN)|$)
Do we need $N$ to be a particular subgroup (like a central subgroup, commutator subgroup, etc) to have a special relationship?
Some thoughts:
Certainly, if $g$ and $hgh^{-1}$ are conjugate elements in $G$, then $(hN)(gN)(hN)^{-1}=hgh^{-1}N$ so $gN$ and $hgh^{-1}N$. The question is if $hgh^{-1}\neq g$ then $hgh^{-1}N=gN$ so $hgh^{-1}g^{-1}\in N$ (So, if $N$ is a derived subgroup then, the conjugacy class all become $1$, i.e. the quotient is abelian. However, what happens in the more general context like if $N$ is a central subgroup like $Z(G)$?)
Second observation if $ccl_G(g)$ is a conjugacy class of size $1$, we can't go any smaller so we know that $|C_{G/N}(gN)|=|C_G(g)|/|N|$