Linear transformations with characteristic subspaces that are not hyperinvariant

If $f$ is an endomorphism of a finite dimensional vector space over a field $K$ then an invariant subspace $X \subseteq V$ is called hyperinvariant (respectively, characteristic) if $X$ is invariant under all endomorphisms (respectively, automorphisms) that commute with $f$. According to Shoda (Math. Zeit. 31, 611--624, 1930) only if $|K| = 2$ then there exist endomorphisms $f$ with invariant subspaces that are characteristic but not hyperinvariant. In this paper we obtain a description of the set of all characteristic non-hyperinvariant subspaces for nilpotent maps $f$ with exactly two unrepeated elementary divisors.