Summary: We prove implications among the conditions in the title for general \(\mathrm{C}^*\)-inclusions \(A\subseteq B\), and we also relate this to several other properties in case \(B\) is a crossed product for an action of a group, inverse semigroup, or an étale groupoid on \(A\). We show that if the \(\mathrm{C}^*\)-inclusion is aperiodic it has a unique pseudo-expectations, and if, in addition, this pseudo-expectation is faithful, then \(A\) supports \(B\) in the sense of the Cuntz preorder. The almost extension property implies aperiodicity, and the converse holds if \(B\) is separable. A crossed product inclusion has the almost extension property if and only if the dual groupoid of the action is topologically principal. Topologically free actions are always aperiodic. If \(A\) is separable or of Type I, then topological freeness, aperiodicity and having a unique pseudo-expectation are equivalent to the condition that \(A\) detects ideals in all \(\mathrm{C}^*\)-algebras \(C\) with \(A\subseteq C\subseteq B\). If, in addition, \(B\) is separable, then all these conditions are equivalent to the almost extension property.