Quotients of $c_0$ that are complemented in $c_0$

Question

Suppose $X$ is a closed subspace of $c_0$ with an unconditional basis and suppose also that it is a quotient of $c_0$. Is $X$ also a complemented subspace of $c_0$?

An affirmative answer implies that $X$ is isomorphic to $c_0$. So, in particular, is every closed subspace of $c_0$ that which is a quotient of $c_0$, isomorphic to $c_0$?

Answer 1

No. $(\sum_{n=1}^\infty \ell_2^n)_{c_0}$ is a quotient of $c_0$.

As I say to my students, "You know this; the problem is to figure out why you know this". :)