  We discuss symmetric quantum measurements and the associated covariant
observables modelled, respectively, as instruments and positive-operator-valued
measures. The emphasis of this work are the optimality properties of the
measurements, namely, extremality, informational completeness, and the rank-1
property which contrast the complementary class of (rank-1) projection-valued
measures. The first half of this work concentrates solely on finite-outcome
measurements symmetric w.r.t. finite groups where we derive exhaustive
characterizations for the pointwise Kraus-operators of covariant instruments
and necessary and sufficient extremality conditions using these
Kraus-operators. We motivate the use of covariance methods by showing that
observables covariant with respect to symmetric groups contain a family of
representatives from both of the complementary optimality classes of
observables and show that even a slight deviation from a rank-1
projection-valued measure can yield an extreme informationally complete rank-1
observable. The latter half of this work derives similar results for continuous
measurements in (possibly) infinite dimensions. As an example we study
covariant phase space instruments, their structure, and extremality properties.
