This question is obviously related to that recent question of mine, but I feel it’s sufficiently different to be posted as a separate question. Let $V$ be a finite-dimensional space. Let ${\cal L}(V)$ denote the space of all endomorphisms of $V$. Say that an endomorphism $\phi$ of ${\cal L}(V)$ is invariant when it satisfies $ \phi (gfg^{-1})=g\phi(f)g^{-1} $ for any $f,g \in {\cal L}(V)$ with $g$ invertible.
Prove or find a counterexample or provide a reference : $\phi$ is invariant iff there are two constants $a,b$ such that $\phi(f)=af+b{\sf tr}(f){\bf id}_V$ for all $f$. With the help of a PARI-GP program, I have checked that this is true when ${\sf dim}(V) \leq 5$. Intuitively, the similitude invariants of a matrix are functions of the coefficients of the characteristic polynomial, and the second largest coefficient, the trace, is the only linear one.