Abelian subgroups of maximum order are isoclinism-invariant

Statement
Suppose $$p$$ is a prime number. Suppose $$P_1$$ and $$P_2$$ are both abelian subgroups of maximum order in $$P$$ that are isoclinic groups. In particular, there is an isomorphism:

$$P_1/Z(P_1) \cong P_2/Z(P_2)$$

Combining with the fourth isomorphism theorem, this isomorphism establishes a bijection between the abelian subgroups of $$P_1$$ containing its center and the abelian subgroups of $$P_2$$ containing its center.

The claim is that under this bijection, the abelian subgroups of maximum order of $$P_1$$ are in bijective correspondence with the abelian subgroups of maximum order of $$P_2$$.

In particular, this means that the index of abelian subgroups of maximum order in $$P_1$$ is the same as the index of abelian subgroups of maximum order in $$P_2$$.