Conjugacy class size statistics need not determine group up to isoclinism

Statement

It is possible to have two finite groups $G_{1}$ and $G_{2}$ such that the multisets of conjugacy class size statistics are the same for $G_{1}$ and $G_{2}$, but such that $G_{1}$ and $G_{2}$ are not isoclinic groups.

Proof

The smallest examples occur for groups of order 32. For more, see Element structure of groups of order 32#Grouping by conjugacy class sizes. Note that the families $\Gamma _{3}$ and $\Gamma _{4}$ have the same conjugacy class size statistics as each other. Similarly, the families $\Gamma _{6}$ and $\Gamma _{7}$ have the same conjugacy class size statistics as each other.

The proof also follows from any of these fact combinations. Note that groups of the same order, the "proportions" of sizes determine the exact sizes:

First fact of pair (specifies an invariant up to isoclinism)	Second fact of pair (notes that the invariant is not determined by degrees of irreducible representations)	Smallest order of an example
isoclinic groups have same proportions of degrees of irreducible representations	conjugacy class size statistics need not determine degrees of irreducible representations	128
isoclinic groups have same nilpotency class	conjugacy class size statistics need not determine nilpotency class	32
isoclinic groups have same derived length	conjugacy class size statistics need not determine derived length	128 (?)