If $A^n$ is similar to $B^n$ for non-nilpotent real metrices A and B, is A similar to B?

Question

let $A,B$ be real non-nilpotent metrices, such that there exists an invertible matrix $P$ for which -

$A^n = P^{-1} B^n P$

holds.

does there always exist an invertible matrix $M$ such that -

$A = M^{-1}BM$ ?

if the answer is yes, does it change if the matrices $A,B$ are over $\mathbb{C}$ ?