  In this note we extend the classical result by G. G. Kasparov that the
Kasparov groups $KK_1(A,B)$ can be identified with the extension groups
$\mbox{Ext}(A,B)$ to the inverse semigroup equivariant setting. More precisely,
we show that $KK_G^1(A,B) \cong \mbox{Ext}_G(A \otimes {\cal K}_G,B \otimes
{\cal K}_G)$ for every countable, $E$-continuous inverse semigroup $G$. For
locally compact second countable groups $G$ this was proved by K. Thomsen, and
technically this note presents an adaption of his proof.
