Isoclinic groups have same proportions of conjugacy class sizes

Statement
Suppose $$G_1$$ and $$G_2$$ are finite groups that are isoclinic groups. Suppose $$c$$ is a positive integer. Denote by $$m_1$$ the number of conjugacy classes of $$G_1$$ of size $$c$$ and denote by $$m_2$$ the number of conjugacy classes of $$G_2$$ of size $$c$$. Then, $$m_1$$ is nonzero if and only if $$m_2$$ is nonzero, and if so, we have:

$$\frac{m_1}{m_2} = \frac{|G_1|}{|G_2|}$$

In other words, given the conjugacy class size statistics of $$G_1$$, we can obtain the conjugacy class size statistics of $$G_2$$ by scaling the number of occurrences of each conjugacy class size by a factor of $$|G_2|/|G_1|$$.

In particular, if $$G_1$$ and $$G_2$$ also have the same order, then they have precisely the same conjugacy class size statistics.

Relation with Schur covering groups
All the Schur covering groups of a given finite group are isoclinic groups, hence have the same conjugacy class size statistics.

Related facts

 * Isoclinic groups have same proportions of degrees of irreducible representations

Facts used

 * 1) uses::Size of conjugacy class equals index of centralizer

Proof outline
The idea behind the proof is to show that the size of the conjugacy class of an element depends only on its coset modulo the center, and is completely determined by the information of the commutator map. We use Fact (1).

Proof details
Given: Two isoclinic groups $$G_1$$ and $$G_2$$, a positive integer $$c$$. $$m_1$$ and $$m_2$$ are respectively the number of conjugacy classes in $$G_1$$ and $$G_2$$ of size $$c$$. Note that the actual number of elements in $$G_1$$ and $$G_2$$ with these conjugacy class sizes are $$m_1c$$ and $$m_2c$$ respectively.

To prove: $$m_1$$ is nonzero if and only if $$m_2$$ is nonzero, and if so, $$\frac{m_1}{m_2} = \frac{|G_1|}{|G_2|}$$

Proof: Let $$W$$ be the group $$\operatorname{Inn}(G_1) \cong \operatorname{Inn}(G_2)$$ and $$T$$ be the group $$G_1' \cong G_2'$$. Denote by $$\gamma: W \times W \to T$$ the map obtained from the commutator map in either group (we know both maps are equivalent via the isoclinism). Denote by $$\alpha_1:G_1 \to W$$ and $$\alpha_2:G_2 \to W$$ the quotient maps modulo the respective centers.