Unlike normality, being characteristic or fully invariant subgroups is
transitive. If
and *A* is characteristic or fully
invariant in *B* and likewise *B* is characteristic or fully invariant in *C*then *A* is characteristic or fully invariant in *C*. By definition an
automorphism of *C* maps *B* to itself (in the characteristic case), and is
thus in turn an automorphism of *B*, which by definition maps *A* to itself.
Similarly an endomorphism of *C* maps a fully invariant *B* into itself, which
is an endomorphism of *B*, which will map *A* into itself.

The reason this does not work for normal subgroups is that while a conjugation
of *C* maps a normal *B* to itself, this mapping of *B* is only known to be an
automorphism of *B*, not a conjugation of *C*, and thus *A* normal in *B* may
not be preserved by the automorphism, and thus not be preserved by the
conjugation of *C*.

For example *Z*_{2} is normal in *D*_{2} which is normal in *D*_{4}, but *Z*_{2} is
not normal in *D*_{4}.