  Let $A$ be a unital separable \CA and $B=C\otimes {\cal K},$ where $C$ is a
unital \CA. Let $\tau: A\to M(B)/B$ be a weakly unital full essential
extensions of $A$ by $B.$ We show that there is a bijection between a quotient
group of $K_0(B)$ onto the set of strong unitary equivalence classes of weakly
unital full essential extensions $\sigma$ such that $[\sigma]=[\tau]$ in
$KK^1(A, B).$ Consequently, when this group is zero, unitarily equivalent full
essential extensions are strongly unitarily equivalent. When $B$ is a
non-unital but $\sigma$-unital simple \CA with continuous scale, we also study
the problem when two approximately unitarily equivalent essential extensions
are strongly approximately unitarily equivalent. A group is used to compute the
strongly approximate unitary equivalence classes in the same approximate
unitary equivalent class of essential
