Let B B denote a compact semigroup with identity and G G a compact abelian group. Let Ext ⁡ ( B , G ) \operatorname {Ext} (B,G) denote the semigroup of extensions of G G by B B . We show that Ext ⁡ ( B , G ) \operatorname {Ext} (B,G) is always a union of groups. We show that it is a semilattice whenever B B is. In case B B is also an abelian inverse semigroup with its subspace of idempotent elements totally disconnected, we obtain a determination of the maximal groups of a commutative version of Ext ⁡ ( B , G ) \operatorname {Ext} (B,G) in terms of the extension functor of discrete abelian groups.