Lipschitz extensions of linear operators

Let $E, F, E_0$ be Banach spaces, with $E_0$ a subspace of $E$. For a maximal Banach operator ideal $\mathcal{A}$, we show that a linear operator from $E_0$ to $F$ can be extended to a linear operator from $E$ to $F$ that belongs to $\mathcal{A}$ if and only if it can be extended to a Lipschitz map from $E$ to $F$ belonging to a wide class of Lipschitz Banach operator ideals related with $\mathcal{A}$. As a consequence, we show that linear operators with special Lipschitz factorization through $\ell_{\infty}(\Gamma)$ has analogous linear factorization through $\ell_{\infty}(\Gamma)$.