  Tate cohomology (as well as Borel homology and cohomology) of connective
K-theory for $G=(\mathbb{Z}/2)^n$ was completely calculated by Bruner and
Greenlees. In this note, we essentially redo the calculation by a different,
more elementary method, and we extend it to $p>2$ prime. We also identify the
resulting spectra, which are products of Eilenberg-Mac Lane spectra, and
finitely many finite Postnikov towers. For $p=2$, we also reconcile our answer
completely with the result of Bruner and Greenlees, which is in a different
form, and hence the comparison involves some non-trivial combinatorics.
