  We study positive kernels on $X\times X$, where $X$ is a set equipped with an
action of a group, and taking values in the set of $\mathcal A$-sesquilinear
forms on a (not necessarily Hilbert) module over a $C^*$-algebra $\mathcal A$.
These maps are assumed to be covariant with respect to the group action on $X$
and a representation of the group in the set of invertible ($\mathcal
A$-linear) module maps. We find necessary and sufficient conditions for
extremality of such kernels in certain convex subsets of positive covariant
kernels. Our focus is mainly on a particular example of these kernels: a
completely positive (CP) covariant map for which we obtain a covariant minimal
dilation (or KSGNS construction). We determine the extreme points of the set of
normalized covariant CP maps and, as a special case, study covariant quantum
observables and instruments whose value space is a transitive space of a
unimodular type-I group. As an example, we discuss the case of instruments that
are covariant with respect to a square-integrable representation.
