  We study the irreducible components of special loci of curves whose group of
symmetries is given as certain group extension. We introduce some relative
Hurwitz data, which we show by using mixed \'etale cohomology theory,
identifies some irreducible components for rational and normal non-abelian
special loci and Hurwitz spaces. A heuristic, that is supported by three
classes of examples, provides an additional context for building further
irreducible loci.
