Transitivity

Unlike normality, being characteristic or fully invariant subgroups is transitive. If $A\subset B\subset C$ and A is characteristic or fully invariant in B and likewise B is characteristic or fully invariant in Cthen 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 Z₂ is normal in D₂ which is normal in D₄, but Z₂ is not normal in D₄.