$\mathcal K$ is the compact operator algebra on a $\omega$-dimensional Hilbert space.
$M$ denotes the multiplier algebra. $Q(B)=M(B\otimes\mathcal K)/B\otimes\mathcal K$.
An extension of $A$ by $B\otimes\mathcal K$ is a short exact sequence:
$0\to B\otimes\mathcal K\to E\to A\to 0$.
Since there is a canonical homomorphism $A=E/B\otimes \mathcal K\to M(B\otimes\mathcal K)/B\otimes\mathcal K=Q(B)$, every extension determines a homomorphism $A\to Q(B)$.
Does every homomorphism $A\to Q(B)$ determine a extension? Or at least, corresponds to an extension?
If $B\otimes \mathcal K$ is essential so that $\varphi:A\to Q(B)$ is injective, $E$ can be chosen to be $\pi^{-1}(\varphi(A))$ in $M(B\otimes\mathcal K)$, and homomorphism $E\to A$ is $\varphi^{-1}\circ \pi$.
Also, if $\varphi:A\to Q(B)$ is $0$, every element of $E$ acts on $B\otimes\mathcal K$ like an element in $B\otimes\mathcal K$. This shows $E=B\otimes\mathcal K\oplus A$, by mapping $x$ to $(\lim xe_\lambda,x+B\otimes\mathcal K)$, where $e_\lambda$ is the approximate unit of $B\otimes\mathcal K$.
What if $\varphi$ is not injective? How do I construct $E$?
(I'm am going to assume $B$ is stable, solely to avoid writing $B\otimes\mathcal K$ so many times. The results hold in the non-stable case, just by tensoring with $\mathcal K$ in the appropriate places.)
Each $*$-homomorphism $\varphi:A\to Q(B)$ determines (an equivalence class of) extensions of $A$ by $B$. Indeed, let $\pi:M(B)\to Q(B)$ denote the quotient map, and define the $C^*$-algebra $$E(\varphi)=\{(a,x)\in A\oplus M(B):\varphi(a)=\pi(x)\}.$$ and define a $*$-homomorphism $q:E(\varphi)\to A$ by $q(a,x)=a$. Then $q$ is surjective, and the kernel of $q$ is $0\oplus B\cong B$. Thus we obtain a short exact sequence $$0\to B\to E(\varphi)\overset{q}{\to} A\to 0.$$
Moreover, if $\eta:0\to B\to E\to A\to 0$ is an extension of $A$ by $B$ and $\varphi:A\to Q(B)$ is its Busby invariant, then the extension $E(\varphi)$ of $A$ by $B$ is isomorphic to $\eta$, in the sense that there is a $*$-isomorphism $E\to E(\varphi)$ making the following diagram commute: