  We prove implications among the conditions in the title for an inclusion of a
C*-algebra A in a C*-algebra 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 \'etale groupoid on A. We show that an aperiodic C*-inclusion
has a unique pseudo-expectation. If, in addition, the unique 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
intermediate C*-algebras. If, in addition, B is separable, then all these
conditions are equivalent to the almost extension property.
