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 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.