Let $C_k$ be the cyclic group of order $k$. Let $G = C_n \rtimes_s C_m$ be the metacyclic group generated by $x$ of order $m$, $y$ of order $n$, such that $x^{-1}yx = y^s$, where $s^m \equiv 1 \pmod n$. Suppose that ${\rm ord}_n(s) < m$.
My question is: Is it always possible to somehow rewrite $G \simeq C_u \rtimes_r C_v$ in such way that ${\rm ord}_u(r) = v$?
I have found the following example: Suppose that $n$ is odd, $m=4$ and $s=-1$, so that ${\rm ord}_n(s) = 2$. Then $C_n \rtimes_{-1} C_4 = \langle x,y \rangle$ is isomorphic to $C_{2n} \rtimes_{-1} C_2 = \langle x,x^2y \rangle$.
However, I don't know how to work with the general case.