The $p$-rank of $\operatorname{Ext}_\mathbb Z(G,\mathbb Z)$ in certain models of $ZFC$

We prove that if the existence of a supercompact cardinal is consistent with $ZFC$, then it is consistent with $ZFC$ that the $p$-rank of $\operatorname{Ext}_\mathbb Z(G,\mathbb Z)$ is as large as possible for every prime $p$ and for any torsion-free Abelian group $G$. Moreover, given an uncountable strong limit cardinal $\mu$ of countable cofinality and a partition of $\Pi$ (the set of primes) into two disjoint subsets $\Pi_0$ and $\Pi_1$, we show that in some model which is very close to $ZFC$, there is an almost free Abelian group $G$ of size $2^\mu=\mu^+$ such that the $p$-rank of $\operatorname{Ext}_\mathbb Z(G,\mathbb Z)$ equals $2^\mu=\mu^+$ for every $p\in\Pi_0$ and 0 otherwise, that is, for $p\in\Pi_1$.