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. 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: