The talk will be devoted to automorphic properties of Banach spaces, in particular \(\ell_\infty/c_0\). A Banach space \(X\) is \(Y\)-automorphic if any isomorphism between two copies of \(Y\) in \(X\) of the same codimension can be extended to an automorphism of \(X\). Intuitively, it means that all embeddings of \(Y\) into \(X\) are the same and cannot be distinguished from the isomorphic point of view. Banach spaces \(c_0(\kappa)\) and \(\ell_2(\kappa)\) are the only known examples of spaces that are automorphic, i.e. \(Y\)-automorphic for every Banach space \(Y\). Space \(\ell_\infty/c_0\) is not automorphic, however, it is separably automorphic (that is, \(Y\)-automorphic for all separable \(Y\)). This last notion has recently been investigated in the context of a related notion of separable injectivity. We are interested in determining the extent of automorphic properties of \(\ell_\infty/c_0\), in particular if they already fail for some subspace of the least uncountable density \(\omega_1\). During the talk, I will present some results on this topic, most of which are included in a joint work with Piotr Koszmider "Almost disjoint families and some automorphic and injective properties of \(\ell_\infty/c_0\)". On one hand, using almost disjoint families we consistently find two copies of \(c_0(\omega_1)\) in \(\ell_\infty/c_0\) with an isomorphism which does not admit an automorphic extension. On the other hand, under Martin's axiom we build such extensions for isomorphisms between certain subspaces, including all copies of \(c_0(\omega_1)\). I will also briefly discuss some related problems and results.