  We analyze the mapping class group of extendible automorphisms of the
exterior boundary W of a compression body of dimension 3 or 4, which extend
over the compression body (Q,V), where V is the interior boundary. Those that
extend as automorphisms of (Q,V) rel V are called discrepant automorphisms,
forming the mapping class group of discrepant automorphisms of W in Q. We
describe a short exact sequence of these mapping class groups. For an
orientable, compact, reducible 3 manifold W, there is a canonical "maximal"
4-dimensional compression body Q whose exterior boundary is W and whose
interior boundary is the disjoint union of the irreducible summands of W. We
obtain a short exact sequence for the mapping class group of a 3-manifold,
which gives the mapping class group of the disjoint union of irreducible
summands as a quotient of the entire mapping class group by the group of
adjusting automorphisms. The group of discrepant automorphisms is described in
terms of generators. The results are useful in a program for classifying
automorphisms of compact 3-manifolds in the spirit of Nielsen-Thurston.
