One of the possible way to define a sum of two extension is:
Let $\mathcal{B}$ and $\mathcal{A}$ be two separable C*-algebras with $\mathcal{B}$ stable and $\phi,\psi: \mathcal{A} \rightarrow \mathcal{M(B)/B}$ be two extensions of $\mathcal{B}$ by $\mathcal{A}$. Then the sum of extension is defined in the following way: $$(\phi \oplus \psi)(a)=\pi(S)\phi(a)\pi(S^*)+\pi(T)\psi(a)\pi(T^*)$$ where $S,T \in \mathcal{M(B)}$ are isometries such that $SS^*+TT^*=1_{\mathcal{M(B)}}$.
The condition $SS^*+TT^*=1_{\mathcal{M(B)}}$ makes sure that the defined sum is still a *-homomorphism. I wonder if there is some deeper connection between isometries and *-homomorphisms in general. I guess what I am asking myself here is how would I guess that to end up with a *-homomorphism I need isometries here. It feels like there is more behind this then I see right now. Thanks for any help!