  We set up a general framework to study Tate cohomology groups of Galois
modules along $\mathbb{Z}_p$-extensions of number fields. Under suitable
assumptions on the Galois modules, we establish the existence of a five-term
exact sequence in a certain quotient category whose objects are simultaneously
direct and inverse systems, subject to some compatibility. The exact sequence
allows one, in particular, to control the behaviour of the Tate cohomology
groups of the units along $\mathbb{Z}_p$-extensions. As an application, we
study the growth of class numbers along what we call "fake
$\mathbb{Z}_p$-extensions of dihedral type". This study relies on a previous
work, where we established a class number formula for dihedral extensions in
terms of the cohomology groups of the units.
