  For most positive integer pairs $(a,b)$, the topological space $#a{\mathbb C
\mathbb P}^2#b{\bar{\mathbb C \mathbb P^2}}$ is shown to admit infinitely many
inequivalent smooth structures which dissolve upon performing a single
connected sum with $S^2\times S^2$. This is then used to construct infinitely
many non-equivalent smooth free actions of suitable finite groups on the
connected sum $#a{\mathbb C \mathbb P}^2#b{\bar{\mathbb C \mathbb P^2}}$. We
then investigate the behavior of the sign of the Yamabe invariant for the
resulting finite covers, and observe that these constructions provide many new
counter-examples to the $4$-dimensional Rosenberg Conjecture.
