  In a previous work, the author established a nonautonomous Conley index based
on the interplay between a nonautonomous evolution operator and its
skew-product formulation. This index is refined to obtain a Conley index for
families of nonautonomous evolution operators. Different variants such as a
categorial index, a homotopy index and a homology index are obtained.
Furthermore, attractor-repeller decompositions and conecting homomorphisms are
introduced for the nonautonomous setting.
