[Part I, cf. the author and \textit{C. Vinsonhaler}, ibid. 39, 195-215 (1982; Zbl 0496.20041).] Let \({\mathcal C}\) be the category of torsion free abelian groups of finite rank, \(G\in {\mathcal C}\), \(C_ 0(G)=\{A\in {\mathcal C}|\) G is projective with respect to all \({\mathcal C}\) exact sequences (pure exact sequences) of the form: \(0\to K\to A\to A/K\to 0\}, P_ 0C_ 0(G)=\{Y\in {\mathcal C}|\) Y is projective with respect to all \({\mathcal C}\) exact sequences: \(0\to K\to A\to A/K\to 0, \forall A\in C_ 0(G)\}\). For \(X=\{X_{\alpha}\}\), a set of groups, \(C_ 0(X)=\cap_{\alpha}C_ 0(X_{\alpha})\) and \(P_ 0C_ 0(X)=\{Y\in {\mathcal C}|\) Y is projective with respect to all \(0\to K\to A\to A/K\to 0, \forall A\in {\mathcal C}_ 0(X)\}\). Let \(G\in {\mathcal C}\) and let \({\mathcal S}_ G\) be the set of all rank one factors of G. The cotypeset of G is the set of all types of torsion free rank one factors of G. Theorem 1. Let G be a torsion free abelian group of finite rank whose cotypeset contains only idempotent types. Then \(P_ 0C_ 0(G)=P_ 0C_ 0({\mathcal S}_ G)\). - If \(X=\{X_{\alpha}\}\) is a set of groups in \({\mathcal C}\) let \({\mathcal I}(X)=\{W\in {\mathcal C}|\quad Ext(X_{\alpha},W)=(0),\quad\forall X_{\alpha}\in X\}\) and let \({\mathcal P}{\mathcal I}(X)=\{Y\in {\mathcal C}| Ext(Y,W)=(0),\quad\forall W\in {\mathcal I}(X)\}. {\mathcal P}{\mathcal I}(X)\) is called the cotorsion-free class of X in \({\mathcal C}\). (It satisfies the requirements for an abstract cotorsion free class.) Let \(\Pi\) be the set of all primes, S an arbitrary non-empty subset of \(\Pi\) and \(\tau\) an arbitrary type. Write \(\tau_{\tilde S}\bar O\) if there exists \(h\in\tau \) such that \(h(p)=0\) for all \(p\in S\). Here \(\bar O=type Z\). If \(\{\tau_{\alpha}\}\) is a set of types let \(\{\tau_{\alpha}\}^ 0=\{S\subseteq\Pi | \tau_{\alpha\tilde S}\bar O\), \(\forall\alpha \}\), \(\{ {\bar\tau }{}_{\alpha}\}=\{types \tau| \tau_{\tilde S}\bar O\), \(\forall S\in\{\tau_{\alpha}\}^ 0\}.\) Theorem 2. Let \(X=\{X_{\alpha}\}\) be a set of groups in \({\mathcal C}\). Then, for \(Y\in {\mathcal C}\), \(Y\in {\mathcal P}{\mathcal I}(X)\) iff \(OT(Y)\in\{OT(X_{\alpha})\}\). - Corollary. Let \(G\in {\mathcal C}\). Then \({\mathcal P}{\mathcal I}(G)={\mathcal P}{\mathcal I}({\mathcal S}_ G)\). - Theorem 3. Let \(G\in {\mathcal C}\). Then \(P_ 0C_ 0\subseteq {\mathcal P}{\mathcal I}(G)\). - Let cotypeset \({}^*G=\{t\in\cot ypeset G| t>\bar O\}\), \([\tau]=\{p\in\Pi | h(p)=\infty\}\). Theorem 4. Let \(G\in {\mathcal C}\). Then \(P_ 0C_ 0(G)={\mathcal P}{\mathcal I}(G)\) iff there exists an idempotent type \(\tau\) with [\(\tau]\) finite such that cotypeset \(^*G=\{\)types \(\tau'|\) \(\bar O<\tau'\leq\tau\}\). - Remark. In this case \(\tau =OT(G)\) and cotypeset G is necessarily finite.