Assuming that two ergodic automorphisms T and R are disjoint, we prove that an ergodic extension $\tilde{T}$ of T remains disjoint with R when, in a representation of $\tilde{T}$ as a skew product over T, the fiber automorphisms belong to a mildly mixing action of a second countable locally countable abelian (LCA) group and the corresponding cocycle with values in this group is ergodic. This allows us to show that the class of ergodic automorphisms which are disjoint from all weakly mixing automorphisms is not closed under taking ergodic joinings, thereby solving a problem stated by E. Glasner and B. Weiss.