Is there a standard term to designate the equivalence class of endomorphisms where two endomorphisms $\phi$ and $\psi$ are considered equivalent if there exists an automorphism $\alpha$ such that $\phi\alpha = \alpha\psi$? Is there some theory about such equivalence classes?
Example. If $M$ is a free module over a commutative ring $R$ with a finite basis of size $n$, and $f$ is an endomorphism of $M$, then the determinant of $f$ can be defined by looking at the endomorphism $$ \bigwedge\nolimits^nM\stackrel{\bigwedge\nolimits^nf}{\longrightarrow}\bigwedge\nolimits^nM $$ and canonically identifying it with an element of $R$, using the fact that the $R$-module $\bigwedge\nolimits^nM$ is free with a $1$-element basis. This definition of the determinant of $f$ can be generalized to obtain other invariants of $f$ by looking at the induced endomorphisms $$ \bigwedge\nolimits^kM\stackrel{\bigwedge\nolimits^kf}{\longrightarrow}\bigwedge\nolimits^kM $$ for all $k$, and taking their equivalence classes up to conjugation by an automorphism.