  The semigroups of unital extensions of separable $C^\ast$-algebras come in
two flavours: a strong and a weak version. By the unital $\mathrm{Ext}$-groups,
we mean the groups of invertible elements in these semigroups. We use the
unital $\mathrm{Ext}$-groups to obtain $K$-theoretic classification of both
unital and non-unital extensions of $C^\ast$-algebras, and in particular we
obtain a complete $K$-theoretic classification of full extensions of UCT
Kirchberg algebras by stable AF algebras.
