Given a (closed) separable subspace of $\ell_\infty$, I am interested in conditions implying that the quotient $\ell_\infty/M$ is isomorphic to a subspace of $\ell_\infty$.

It is not difficult to see that being $M$ reflexive is sufficient, and Bourgain proved that $\ell_\infty/c_0$ does not admit an equivalent strictly convex norm (while $\ell_\infty$ does). So $\ell_\infty/c_0$ is not isomorphic to a subspace of $\ell_\infty$.

Given a (closed) separable subspace $M$ of $\ell_\infty$, I am interested in conditions implying that the quotient $\ell_\infty/M$ is isomorphic to a subspace of $\ell_\infty$.

It is not difficult to see that being $M$ reflexive is sufficient, and Bourgain proved that $\ell_\infty/c_0$ does not admit an equivalent strictly convex norm (while $\ell_\infty$ does). So $\ell_\infty/c_0$ is not isomorphic to a subspace of $\ell_\infty$.