Forcing a Basis into $\aleph_1$-Free Groups

In this paper, we address the question of when a non-free $\aleph_1$-free group $H$ can be be free in a transitive cardinality-preserving model extension. Using the $\Gamma$-invariant, denoted $\Gamma(H)$, we present a necessary and sufficient condition resolving this question for $\aleph_1$-free groups of cardinality $\aleph_1$. Specifically, if $\Gamma(H) = [\aleph_1]$, then $H$ will be free in a transitive model extension if and only if $\aleph_1$ collapses, while for $\Gamma(H) \ne [\aleph_1]$ there exist cardinality-preserving forcings that will add a basis to $H$. In particular, for $\Gamma(H) \neq [\aleph_1]$, we provide a poset $(\mathcal P_{\rm pb}, \leq)$ of partial bases for adding a basis to $H$ without collapsing $\aleph_1$.